document: this article is motivated by the following result of berestycki , et al . given in @xcite for the laplacian perturbed by a divergence - free drift in dimensions @xmath10 . 
   let @xmath11 be a bounded @xmath12 regular open set and let @xmath13 be a bounded @xmath7-dimensional vector field such that @xmath14 on @xmath2 in the sense of distributions ( distr . ) , i.e. @xmath15 for @xmath5 , let ( @xmath16 ) be the principal eigen - pair corresponding to the dirichlet problem for the operator @xmath17 . 
   theorem 0.3 of @xcite asserts that @xmath18 remains bounded as @xmath9 , if and only if the equation @xmath19 has a solution @xmath20 ( called a first integral of @xmath6 ) , such that @xmath21 and @xmath22 
   .    the result can be interpreted intuitively in the following way : functions @xmath20 satisfying are constant along the flow of the vector field @xmath23 ( see section  [ sec : ub ] ) , and the existence of ( non - trivial ) first integrals allows for flow lines that are contained in @xmath2 . on the other hand , if no such @xmath20 exist , then the flow leaves @xmath2 with speed proportional to @xmath24 . adding the laplacian @xmath25 to @xmath26 , or equivalently the brownian motion to the flow , results in a stochastic process whose trajectories gradually depart from the integral curves of @xmath6 , but the general picture is similar : 
   if nontrivial first integrals exist , then the trajectories may remain in @xmath2 with positive probability during a finite time interval , even as @xmath9 . in this case 
   we are lead to a nontrivial limiting transition mechanism between the flow lines . 
   the result described in the foregoing enjoys many extensions and has proved quite useful in various applications describing the influence of a fluid flow on a diffusion , see for example @xcite . in the context of a compact , connected riemannian manifold a sufficient and necessary condition for @xmath18 to remain bounded , as @xmath9 , expressed in terms of the eigenspaces of the advection operator @xmath27 , has been given in ( * ? ? ? 
   * theorem 1 ) . 
   the purpose of the present paper is to verify that a similar property of the principal eigenvalue holds when the classical laplacian is replaced by the fractional laplacian @xmath4 with @xmath28 . 
   we consider @xmath29 defined as the set of all the nonzero first integrals in the sobolev space @xmath30 equipped with the norm coming from the dirichlet form @xmath31 of @xmath4 ( see below ) . 
   the sobolev norm condition on the first integrals reflects smoothing properties of the green function of the fractional laplacian , while is related to the flow defined by @xmath6 . 
   the main difficulty in our development stems from roughness of general elements of @xmath32 and non - locality of @xmath4 , which prevent us from a direct application of the differential calculus in the way it has been done in @xcite . 
   instead , we use conditioning suggested by a paper of bogdan and dyda @xcite , approximation techniques for flows given by diperna and lions in @xcite , and the properties of the green function and heat kernel of gradient perturbations of @xmath4 obtained by bogdan , jakubowski in @xcite and chen , et al . in @xcite for @xmath3 and bounded @xmath1-regular open sets @xmath2 . these properties allow to define and study , via the classical krein - rutman theorem and compactness arguments , the principal eigen - pair @xmath33 for @xmath34 and @xmath3 . 
   our main result can be stated as follows . 
   [ main - thm ] suppose that @xmath35 is a bounded domain with @xmath1-regular boundary that is simply connected , i.e. @xmath36 - the complement of @xmath37 - is connected . 
   if @xmath28 , and @xmath38 is of zero divergence , then @xmath39 and the infimum is attained . 
   here we use the convention that @xmath40 , hence @xmath41 if and only if the zero function is the only first integral . 
   equality results from the following lower and upper bounds of @xmath18 , @xmath42 @xmath43 the bounds are proved in sections  [ sec : lb ] and [ sec : ub ] , correspondingly . in section [ proof - coro ] 
   we explain that the minimum on the right hand side of is attained , and we finish the proof of the theorem .    comparing our approach with the arguments used in the case of local operators , cf . 
   @xcite , we note that the use of the green function seems more robust whenever we lack sufficient differentiability of functions appearing in variational formulas . recall that in the present case we need to deal with @xmath30 , which limits the applicability of the arguments based on the usual differentiation rules of the classical calculus , e.g. the leibnitz formula or the chain rule . 
   we consider the use of the green function as one of the major features of our approach . 
   in addition , the non - locality of the quadratic forms forces a substantial modifications of several other arguments , e.g. those involving conditioning of nonlocal operators and quadratic forms in the proof of the upper bound in section [ sec : ub ] . 
   finally , we stress the fact that the dirichlet fractional laplacian on a bounded domain @xmath2 _ is not _ a fractional power of the dirichlet laplacian on @xmath2 , e.g. the eigenfunctions of these operators have a different power - type decay at the boundary , see @xcite in this connection .    as a preparation for the proof , we recall in section [ sec : p ] the estimates of @xcite for the green function and transition density of @xmath44 for the dirichlet problem on @xmath2 . 
   these functions are defined using hunt s formula , which in principle requires the drift @xmath45 to be defined on the entire @xmath46 . 
   we show however , in corollary [ cor010212 ] , that they are determined by the restriction of the drift to the domain @xmath2 . in section [ sec3 ] 
   we prove that the corresponding green s and transition operators are compact , see lemmas [ lem : gdc1 ] and [ lem : gdc1 kb ] . 
   this result is used to define the principal eigen - pair of @xmath44 , via the krein - rutman theorem . in theorem [ thm011307 ] of section [ sec4 ] 
   we prove that the domains of @xmath4 and @xmath44 in @xmath47 coincide . in section 
   [ sec : pt11 ] we employ the bilinear form of @xmath44 to estimate the principal eigenvalue . 
   the technical assumption @xmath48 is only needed in sections  [ sec : ub ] to characterize the first integrals of @xmath6 by means of the theory of flows developed by diperna and lions in @xcite for sobolev - regular vector fields . 
   we start with a brief description of the setting and recapitulation of some of the results of @xcite . 
   further details and references may be found in those papers ( see also @xcite and the references therein ) . in what follows , @xmath49 is the euclidean space of dimension @xmath50 , scalar product @xmath51 , norm @xmath52 and lebesgue measure @xmath53 . 
   all sets , measures and functions in @xmath49 considered throughout this paper will be borel . 
   we denote by @xmath54 the ball of center @xmath55 and radius @xmath56 . 
   we will consider nonempty , bounded open set @xmath35 , whose boundary is of class @xmath1 . 
   the latter means that @xmath56 exists such that for every @xmath57 there are balls @xmath58 and @xmath59 , which are tangent at @xmath60 ( the _ inner _ and _ outer _ tangent ball , respectively ) . 
   we will refer to such sets @xmath2 as to @xmath1 domains , without requiring connectivity . for an alternative analytic description and localization of @xmath1 domains we refer to ( * ? ? ? 
   * lemma  1 ) . 
   we note that each connected component of @xmath2 contains a ball of radius @xmath61 , and the same is true for @xmath62 . 
   therefore @xmath2 and @xmath62 have a finite number of components , which will play a role in a later discussion of extensions of the vector field to a neighborhood of @xmath63 . 
   the distance of a given @xmath55 to @xmath62 will be denoted by @xmath64    _ constants _ mean positive numbers , that do not depend on the considered arguments of the functions being compared . 
   accordingly , notation @xmath65 means that there is a constant @xmath66 such that @xmath67 for all @xmath68 . as usual , @xmath69 and @xmath70 . 
   we will employ the function space @xmath47 , consisting of all square integrable real valued functions , with the usual scalar product @xmath71 generally , given @xmath72 , the norms in @xmath73 shall be denoted by @xmath74 . 
   customarily , @xmath75 denotes the space of smooth functions on @xmath49 with compact support in @xmath2 . 
   also , @xmath76 denotes the closure of @xmath75 in the norm @xmath77 , where @xmath78 in the last equality we have used plancherel theorem . the fourier transform of @xmath79 is given by @xmath80      the discussion in section  [ sec : iso ] and section  [ sec : kill ] is valid for any @xmath82 . 
   let @xmath83 the coefficient is chosen in such a way that @xmath84\nu(y)dy=|\xi|^\alpha\,,\quad    \xi\in \rd\,.\ ] ] we define the _ fractional laplacian _ as the @xmath85-closure of the operator @xmath86\nu(y)dy\ , ,    \quad    x\in \rd,\,\phi\in c^\infty_c(\rd).\ ] ] its fourier symbol is given by @xmath87 , cf . 
   the fractional laplacian is the generator of the semigroup of the isotropic @xmath81-stable lvy process @xmath88 on @xmath49 . here 
   @xmath89 and @xmath90 are the law and expectation for the process starting at @xmath91 . 
   these are defined on the borel @xmath92-algebra of the canonical cdlg path space @xmath93 via transition probability densities as follows . 
   we let @xmath94 be the canonical process , i.e. @xmath95 and define time - homogeneous transition density @xmath96 where @xmath97 according to ( [ eq : trf ] ) and the lvy - khinchine formula , @xmath98 is a probabilistic convolution semigroup of functions with the lvy measure @xmath99 , see e.g. @xcite . 
   from ( [ eq : dpt ] ) we have @xmath100 it is well - known ( @xcite ) that @xmath101 , hence @xmath102      let @xmath103 be the _ time of the first exit _ of the ( canonical ) process from @xmath2 . for each @xmath104 , the measure @xmath105\ ] ] is absolutely continuous with respect to the lebesgue measure on @xmath2 . 
   its density @xmath106 is continuous in @xmath107 and satisfies g.  hunt s formula ( see @xcite , @xcite ) , @xmath108.\ ] ] in addition , the kernel is symmetric ( see @xcite for discussion and references ) : @xmath109 and defines a strongly continuous semigroup on @xmath47 , @xmath110 we shall denote @xmath111 . 
   the _ green function _ of @xmath4 for @xmath2 is defined as @xmath112 and the respective _ green operator _ is @xmath113 = \int_{\rd}g_d(x , y)f(y)dy.\end{aligned}\ ] ] we have ( @xcite ) , @xmath114 and @xmath115 see ( * ? ? ? 
   * lemma  5.3 ) , where the fractional laplacian operator appearing above is defined in the sense of distributions theory . 
   see formula below for another statement . 
   thus the generator of @xmath116 is @xmath117 with zero ( dirichlet ) exterior conditions , i.e. with the domain equal to the range of @xmath118 , @xmath119 the following estimate has been proved by kulczycki @xcite and chen and song @xcite ( see also ( * ? ? ? 
   * theorem 21 ) ) ,    @xmath120^\alpha}\label{greenestimatestj}\\ & &   \approx\ ;   |x - y|^{\alpha - d}\left(\frac{\delta_d(x)^{\alpha/2}\delta_d(y)^{\alpha/2}}{|x - y|^\alpha } \land 1\right)\nonumber \ , , \qquad x , y \in d\,.\nonumber\end{aligned}\ ] ]    in particular , @xmath121 from it also follows that @xmath122 from ( * ? ? ? 
   * corollary 3.3 ) we have the following gradient estimate , @xmath123 we define @xmath124 as the subspace of @xmath125 made of those elements for which @xmath126 is given as follows ( cf @xcite ) : @xmath127 ^ 2\nu(y - x)dxdy,\label{011109b}\end{aligned}\ ] ] the above formula can be also used to define a bilinear form @xmath128 on @xmath129 by polarization . 
   we also have @xmath130 therefore @xmath131 we define @xmath32 as the closure of @xmath75 in the norm @xmath132 . by theorems 4.4.2 , a.2.10 and formula ( 4.3.1 ) of @xcite , we have ( cf . ) ) , @xmath133 if @xmath134 and @xmath135 belongs to the domain of the fractional laplacian , then @xmath136      throughout the remainder of the paper we always assume that @xmath137 , and @xmath138 is a bounded vector field . 
   for @xmath139 and @xmath140 we let @xmath141 and for each @xmath142 @xmath143    let @xmath144 it follows from ( * ? ? ? 
   * theorem  2 and example  2 ) that series converges uniformly on compact subsets of @xmath145 . from the results of @xcite 
   , we know that @xmath146 is a transition probability density function , i.e. it is non - negative and @xmath147 in addition , @xmath148 is continuous on @xmath145 , and @xmath149 where @xmath150 if @xmath151 . 
   in fact , this holds under much weaker , kato - type condition on @xmath6 , see ( * ? ? ? 
   * theorems  1 and 2 ) . 
   we denote by @xmath152 and @xmath153 the law and expectation on @xmath154 for the ( canonical ) markov process starting at @xmath68 and defined by the transition probability density @xmath155 ,    @xmath152 may also be defined by solving stochastic differential equation @xmath156 . 
   such equations have been studied in dimension @xmath157 in @xcite under the assumptions of boundedness and continuity of the vector field ; also for @xmath158 . 
   we refer the reader to ( * ? ? ? 
   * formula ( 13 ) ) , for a closer description of a connection to and . 
   we may also define the perturbation series for @xmath159 . in 
   what follows , objects pertaining to @xmath160 will be marked with the superscript hash ( @xmath161 ) , e.g. @xmath162      if @xmath163 on @xmath2 in the sense of distributions theory , and @xmath164 , then @xmath165 
   if also @xmath166 and we substitute @xmath167 for @xmath79 in , then we obtain @xmath168 the last equality extends to arbitrary @xmath169 .    [ lem : duality ] if @xmath163 on @xmath49 , then we have @xmath170 and @xmath171 for all @xmath139 , @xmath172 , and @xmath173 . 
   let @xmath174 , @xmath172 . by 
   ( see also ( * ? ? ? 
   * lemma  4 ) ) , @xmath175 from we conclude that @xmath176 . for a general @xmath177 , by we have @xmath178 where @xmath179 , @xmath180 , @xmath181,\ ] ] and @xmath182,\ ] ] with the convention that @xmath183 , cf . . 
   using formula @xmath184 times in space , and then integrating in time we see that @xmath185 which yields the identities stated in the lemma .    a strengthening of proposition  [ lem : duality ] 
   will be given in corollary  [ cor010908 ] below .    if @xmath163 on @xmath49 , then and proposition [ lem : duality ] 
   yield @xmath186      we recall that @xmath2 is a bounded @xmath1 domain in @xmath49 . 
   hunt s formula may be used to define the transition probability density of the ( first ) perturbed and ( then ) killed process @xcite . 
   thus , for @xmath139 , @xmath187 we let @xmath188\,.\ ] ] we have @xmath189=\int_d \tp_d(t , x , y)f(y)dy,\quad t>0,\,x\in d,\,f\in l^\infty(d),\ ] ] and @xmath190 where @xmath191 } c_t<+\infty$ ] for any @xmath192 .    by ( * ? 
   * formula ( 40 ) ) , there exist constants @xmath193 such that @xmath194    we define the green function of @xmath2 for @xmath195 : @xmath196 clearly @xmath197 is nonnegative . by blumenthal s 0 - 1 law , @xmath198 for all @xmath139 ( and thus @xmath199 ) if @xmath200 or @xmath201 , see ( [ ptxy_comp ] ) . 
   it follows from and ( see ( * ? ? ? 
   * lemma 7 ) ) , that @xmath202 the main result of @xcite asserts that @xmath203 and @xmath204 is continuous for @xmath205 ( for estimates of @xmath206 see @xcite ) . 
   we consider the integral operators @xmath207 and @xmath208 in light of and , the above operators are bounded on every @xmath73 , @xmath209 . 
   the analogous operators @xmath210 and @xmath211 , defined for @xmath159 , turn out to be mutually adjoint on @xmath47 , as follows from corollary [ cor010908 ] below . from , and we obtain that @xmath212 is a semigroup of contractions in every @xmath73 , @xmath213 $ ] . 
   it is strongly continuous for @xmath209 . 
   let @xmath195 be the @xmath47 generator of the semigroup , with the domain @xmath214 . 
   we have @xmath215 it has been shown in @xcite that the following crucial recursive formula holds @xmath216 and for all @xmath217 and @xmath218 we have @xmath219 later on we shall also consider the operator @xmath220 corresponding to the vector field @xmath221 , where @xmath5 and we let @xmath222 . 
   clearly , if @xmath24 is fixed , then there is no loss of generality to focus on @xmath223 . 
   [ sec4 ]    the following pointwise version of is proved in ( * ? ? ? 
   * lemma 12 ) , @xmath224 with @xmath225 we define @xmath226 after a series of auxiliary estimates , we will prove @xmath227 to be compact on @xmath47 . 
   [ lem011107 ] there exist @xmath228 such that @xmath229    using ( [ eq : gradestimgreen ] ) we obtain that @xmath230}{\delta_d(x)|x - y|},\ ] ] which we bound from above , thanks to ( [ greenestimatestj ] ) , by @xmath231\delta_d(y)^{\alpha/2}}{[\delta_d(x)\vee    |x - y| \vee \delta_d(y)]^{\alpha}}\ ,    \nonumber\\ & \le c_1\,\delta_d(x)^{\alpha/2 - 1}|x - y|^{\alpha-1-d } \dfrac{\delta_d(y)^{\alpha/2}}{[\delta_d(x)\vee    |x - y| \vee \delta_d(y)]^{\alpha-1}}\ , ,   \end{aligned}\ ] ] and this yields .    [ lem051307 ] if @xmath232 , then there is @xmath233 such that @xmath234    we only need to examine points close to @xmath235 . 
   given such a point we consider the integral over its neighborhood @xmath236 . 
   the neighborhood can be chosen in such a way that , after a bi - lipschitz change of variables ( see ( * ? ? ? 
   * formula ( 75 ) ) , ( * ? ? ? * formula ( 11 ) ) ) , we can reduce our consideration to the case when @xmath237 $ ] , @xmath238 $ ] and @xmath239 for @xmath240 . 
   the respective integral over @xmath236 is then estimated by @xmath241    the result is valid for all bounded open _ 
   lipschitz _ sets ( @xcite ) in all dimensions @xmath242 . 
   lemma  [ lem051307 ] yields the following .    [ cordk ] let @xmath243 and @xmath244 . 
   suppose also that @xmath245 denote @xmath246 then , @xmath247 constants @xmath248 depend only on @xmath249    the above result shall be used to establish that operator @xmath250 ( thus also @xmath227 ) is @xmath251 bounded via schur s test , see ( * ? ? ? 
   * theorem 5.2 ) . note that we have @xmath252 , for some constant @xmath253 , provided that @xmath254 . 
   likewise , @xmath255 , provided that @xmath256 . 
   summarizing , we will require the following conditions : @xmath257    [ lemary ] conditions ( [ c1])-([c3 ] ) hold , if @xmath258 and @xmath259 where @xmath260 . 
   we have @xmath261 , etc . 
   [ prop021307 ] operator @xmath227 is compact on @xmath47 . 
   we note that @xmath262 , where @xmath263 is given by and @xmath253 is some constant . 
   let @xmath264 be as in lemma  [ lemary ] . 
   by corollary  [ cordk ] and the discussion preceding lemma  [ lemary ] , there are constants @xmath265 such that @xmath266 using schur s test for boundedness of integral operators on @xmath251 spaces ( see ( * ? ? ? * theorem 5.2 ) ) we obtain @xmath267 for @xmath268 , @xmath269 we let @xmath270 , @xmath271 , and @xmath272 each operator @xmath273 is hilbert - schmidt , hence compact ( ( * ? ? ? 
   * theorem 4 , p. 247 ) ) , due to the fact that @xmath274 we have @xmath275 , unless @xmath276 . 
   the latter may only happen if @xmath277 satisfy @xmath278 , with @xmath228 ( cf ) , or equivalently when @xmath279 , and then there are constants @xmath280 such that @xmath281    this estimate for @xmath282 actually holds for all @xmath283 , hence yields @xmath284 . since @xmath227 is a norm limit of compact operators 
   , it is compact , too .    in view of proposition[prop021307 ] we may regard the gradient operator @xmath26 as a small perturbation of @xmath4 when @xmath3 . 
   in fact @xmath26 is relatively compact with respect to @xmath4 with dirichlet conditions in the sense of ( * ? ? ? 
   * iv.1.3 ) . 
   [ thm011307 ] @xmath285 .    by virtue of 
   , we can write @xmath286 by , @xmath118 is injective , therefore so is @xmath287 . in particular 
   , @xmath157 is not an eigenvalue of @xmath227 . since @xmath227 is compact , by the riesz - schauder theory ( * ? ? ? 
   * theorem x.5.1 , p. 283 ) , @xmath287 is invertible . 
   thus , @xmath288 for future reference we remark that @xmath289 is invertible , too . 
   [ cor011307 ] @xmath290 .    by ( 
   * lemma 10 ) , @xmath291 for any bounded function @xmath135 . invoking the argument used in proposition [ prop021307 ] , we conclude that there is a number @xmath292 independent of @xmath135 for which @xmath293 by approximation , and extend to all @xmath294 . 
   furthermore , if @xmath295 and @xmath296 in @xmath47 , then @xmath297 and @xmath298 on @xmath235 , hence @xmath299 ( see ( * ? ? ? 
   * theorem 5.37 ) ) . 
   therefore , @xmath300 . 
   the following result justifies our notation @xmath301 . 
   [ prop011009 ] if @xmath302 , then @xmath303 . 
   let @xmath304 and @xmath305 . 
   applying @xmath306 to @xmath307 and using , we obtain @xmath308 which , thanks to , concludes the proof . 
   we shall also observe the following localization principle for our perturbation problem . 
   [ cor010212 ] operators @xmath306 and @xmath309 do not depend on the values of @xmath6 on @xmath62 .    if @xmath310 and @xmath311 are two ( bounded ) vector fields on @xmath49 equal to @xmath6 on @xmath2 , and @xmath312 and @xmath313 are the corresponding green s operators , then , by , @xmath314 which identifies @xmath315 as operators on @xmath47 , hence also as functions defined in section  [ sec : deftg ] . 
   a similar conclusion for @xmath316 follows from the fact that @xmath317 is the generator of the semigroup . 
   we will identify the adjoint operator of @xmath227 on @xmath47 . 
   [ lemhs ] @xmath318 . 
   if @xmath319 , then by corollary  [ cor011307 ] and , @xmath320 functions @xmath321 form a dense set in @xmath47 , which ends the proof . 
   when defining @xmath322 , via , we may encounter the situation when @xmath6 is given only on @xmath2 . 
   we may extend the field outside @xmath2 by letting e.g. @xmath323 on @xmath62 . 
   of course such an extension needs not satisfy @xmath324 , even though the condition may hold on @xmath2 . 
   however , we still have a local analogue of proposition [ lem : duality ] . 
   recall that the transition probability densities and green function corresponding to the vector field @xmath160 are marked with a hash ( @xmath161 ) . 
   [ cor010908 ] if @xmath325 on @xmath2 , then for all @xmath139 and @xmath283 , @xmath326 and @xmath327    we shall first prove , or , equivalently , that @xmath328 where @xmath329 is adjoint to @xmath330 on @xmath47 . from applied to @xmath331 
   we have , @xmath332 applying the operator @xmath118 from the left to both sides of the equality we obtain @xmath333 by lemma  [ lemhs ] , @xmath334 taking adjoints of both sides of we also obtain , @xmath335 we have already noted in the proof of theorem [ thm011307 ] that @xmath289 is a linear automorphism of @xmath47 
   . therefore and give . 
   furthermore , let @xmath336 denote the adjoint of @xmath195 on @xmath47 . 
   we have @xmath337 where the last equality follows from ( * ? ? ? 
   * lemma xii.1.6 ) . 
   let @xmath338 be the semigroup adjoint to @xmath339 . by ( * ? 
   * corollary 4.3.7 ) , the generator of @xmath338 is @xmath340 . since @xmath341 , the semigroups are equal . 
   the corresponding kernels are defined pointwise , therefore they satisfy . 
   [ lem : gdc1 ] operator @xmath342 is hilbert - schmidt on @xmath47 for each @xmath139 . 
   let @xmath343 be an orthonormal base in @xmath47 . by and plancherel s identity 
   , we bound the hilbert - schmidt norm as follows . using and @xmath344    [ lem : gdc1 kb ] @xmath197 is compact on @xmath47 . 
   let @xmath268 , and @xmath345 . 
   the integral operator @xmath346 on @xmath47 corresponding to the kernel @xmath347 is compact . 
   indeed , it has a finite hilbert - schmidt norm : @xmath348 ^ 2dxdy\le n^2|d|^2<\infty.\ ] ] the norm of @xmath349 on @xmath47 may be directly estimated as follows , @xmath350 see , e.g. , theorem 3 , p. 176 of @xcite . 
   we let @xmath351 . 
   the functions @xmath352 ( and @xmath353 ) are uniformly integrable on @xmath2 by . since @xmath197 is approximated in the norm topology by compact operators , it is compact .    in the special case when @xmath354 , @xmath355 equals @xmath356 , a symmetric contraction semigroup on @xmath47 , whence the green operator @xmath118 is symmetric , compact and positive definite . 
   the spectral theorem yields the following .    [ cor011206 ] @xmath357 is symmetric and compact for every @xmath358 .    by and , @xmath197 is also irreducible . krein - rutman theorem ( see @xcite ) implies that there exists a unique nonnegative @xmath359 and a number @xmath360 such that @xmath361 and @xmath362 we shall call @xmath363 the _ principal eigenpair _ corresponding to @xmath195 . from 
   we have @xmath364 and @xmath365 . 
   furthermore , @xmath366 the formula yields extra regularity of @xmath367 , as follows . 
   [ lem011106 ] if holds for some @xmath368 and @xmath369 , then @xmath370 and there is @xmath371 such that latexmath:[\[\label{011106 }     starting from , for an arbitrary integer @xmath142 we obtain @xmath373 where @xmath374 and @xmath375 to estimate @xmath376 , we use basic properties of the bessel potentials , which can be found in ( * ? ? ? 
   * ch.ii.4 ) . recall that for @xmath377 the bessel potential kernel @xmath378 is the unique , extended - continuous , probability density function on @xmath49 , whose fourier transform is @xmath379 thus , @xmath380 for all @xmath381 , see ( 4.7 ) of ibid . if @xmath382 , then by ( * ? ? ? 
   * ( 4.2 ) ) , @xmath383 is locally comparable with @xmath384 . 
   by there is a constant @xmath253 such that @xmath385 hence , @xmath386 , which is bounded if @xmath387 , see ( * ? ? ? * ( 4.2 ) ) again . 
   considering such @xmath184 we conclude that @xmath367 is bounded . 
   the boundary decay of @xmath367 follows from , and . 
   the continuity of @xmath367 is a consequence of the continuity of @xmath204 for @xmath388 , and the uniform integrability of the kernel , which stems from . 
   we say that @xmath20 is a _ first integral _ of @xmath6 if @xmath389 cf . . 
   we write @xmath390 if @xmath391 and @xmath20 is not equal to @xmath392 a.e . 
   recall that for @xmath5 , the operator @xmath393 is considered with the dirichlet exterior condition on @xmath2 , i.e. it acts on @xmath394 , see theorem  [ thm011307 ] . 
   the green operator and krein - rutman eigen - pair of @xmath44 shall be denoted by @xmath395 and @xmath33 , respectively . 
   we also recall that @xmath396 and @xmath397 the proof of shall be obtained by demonstration of lower and upper bounds for @xmath18 ( as @xmath222 ) . 
   [ prop010906 ] if @xmath398 , then @xmath399 . 
   _ let @xmath398 . according to proposition [ prop011009 ] , @xmath79 belongs to @xmath400 and @xmath401 . 
   in addition , @xmath402 . taking the scalar product of both sides of the equality against @xmath79 and using we get the result , because the second term vanishes .    according to proposition [ prop010906 ] , @xmath403 suppose @xmath404 , as @xmath405 , but @xmath406 stay bounded . by and corollary [ cor011206 ] , 
   the sequence @xmath407 is pre - compact in @xmath47 . 
   suppose that @xmath20 is a weak limit of @xmath408 in @xmath30 , thus a strong limit in @xmath47 . 
   we have @xmath409 , and for @xmath410 , @xmath411 dividing both sides by @xmath412 and passing to the limit , we obtain that @xmath413 thus @xmath414 . 
   fatou s lemma and yield @xmath415 , therefore follows .      the proof of uses `` conditioning '' of truncations of @xmath416 by the principal eigenfunction inspired by @xcite and @xcite ( see below ) . 
   here @xmath20 is a first integral in @xmath32 . 
   an important part of the procedure is to prove that the truncation of @xmath416 is also a first integral in @xmath32 . 
   this is true if @xmath417 : suppose that @xmath418 and @xmath419 are bounded and @xmath420 . 
   let @xmath163 and that @xmath421 a.e . 
   then , a.e . 
   we have @xmath422 thus @xmath423 is also a first integral of @xmath6 . however , the a.e . 
   differentiability of @xmath424 is not guaranteed for @xmath425 . 
   in fact , for @xmath424 , the condition @xmath426 is understood in the sense of distributions theory , cf , and the calculations in may only serve as a motivation . to build up tools for a rigorous proof of the distributional version of , in section [ sec5.2.1 ] for a sobolev - regular , divergence - free vector field @xmath138 , we consider the incompressible flow @xmath427 of mappings on @xmath46 constructed by diperna and lions in ( * ? ? ? 
   * theorem iii.1 ) ( the integral curves of @xmath6 ) . 
   the flow is used in lemma [ lem012707 ] to characterize the first integrals in @xmath428 as those locally invariant under the flow . 
   then , we are able to conclude that the composition of a first integral in @xmath428 with a lipschitz function is also a first integral in @xmath428 , see corollary [ prop011007 ] , and , finally , use the conditioning of @xmath416 .      unless stated otherwise , in this section we consider general @xmath138 such that @xmath429 and @xmath163 a.e . on @xmath46 . 
   according to ( * ? ? ? 
   * theorem iii.1 ) , there exists a unique    a.e . 
   defined jointly borelian family of mappings @xmath430 ( flow generated by @xmath6 ) with the following properties : first , for a.e . 
   @xmath55 , the function @xmath431 is continuous and @xmath432 so that , in particular , @xmath433 second , for all @xmath434 and borel measurable sets @xmath435 , @xmath436 where @xmath437 is the @xmath7-dimensional lebesgue measure , and , third , for all @xmath438 , @xmath439 by and , if @xmath440 are nonnegative , then @xmath441 we let @xmath213 $ ] , @xmath442 , and define @xmath443 note that , @xmath444 , @xmath445 , defines a group of isometries on @xmath446 . if @xmath447 , then @xmath448 for a.e . 
   @xmath55 and all @xmath434 , because @xmath163 implies @xmath449 . 
   then for all @xmath434 , @xmath450 dxds.\end{aligned}\ ] ] let @xmath451 , @xmath452 and @xmath453 ( a mollifier ) . 
   define @xmath454 the function is an approximate solution of the transport equation @xmath455 in the following sense : if @xmath456 @xmath457 $ ] , @xmath458 and @xmath459 , then for all @xmath434 and finite @xmath460 , @xmath461 this follows from ( * ? ? ? 
   * theorem ii.1 ) . 
   we shall use @xmath462 to characterize the first integrals @xmath463 as those elements of @xmath32 , which are constant along the flow @xmath464 . 
   a word of explanation may be helpful . 
   suppose that @xmath465 , @xmath163 a.e . 
   , and @xmath20 is an @xmath466-integrable first integral , i.e. @xmath467 , as distributions on space - time @xmath468 , see ( * ? ? ? * ( 13 ) ) . by ( * ? ? ? 
   * corollary  ii.1 ) , there is a unique solution to the transport equation with the initial condition @xmath469 . 
   the equation is understood in the sense of distributions on space - time , too . by ( * ? ? ? 
   * theorem  iii.1 ) , the solution has the form @xmath470 . 
   however , since @xmath20 is a first integral , the mapping @xmath471 defines another solution . 
   thus , by uniqueness , @xmath472 a.e . in our case 
   this argument needs to be slightly modified since the first integral is defined only on @xmath2 and not on the entire @xmath49 . in particular , the identity @xmath472 is bound to hold only for small times @xmath473 . 
   [ lem012707 ] let @xmath474 , @xmath475 , and @xmath476 
   . then @xmath414 if and only for every @xmath477 there is @xmath478 such that @xmath479    suppose that @xmath480 satisfies with @xmath478 and let @xmath481 . 
   choose @xmath477 so that @xmath482 for all @xmath68 in the support of @xmath483 . using and , for all @xmath484 we obtain , @xmath485 applying , 
   we rewrite the rightmost side of to obtain , @xmath486 as a result , @xmath487 letting @xmath488 we see that @xmath489 the limiting passage is justified by boundedness of @xmath6 , integrability of @xmath20 ( cf . 
   the discussion preceding ) and dominated convergence theorem . since @xmath481 is arbitrary 
   , we conclude that @xmath390 . 
   conversely , let us assume for some @xmath476 . 
   by sobolev embedding theorem ( * ? ? 
   * theorem v.1 , p. 119 ) , we have that @xmath490 , where @xmath491 . 
   let @xmath474 , @xmath492 on @xmath2 and @xmath493 on @xmath62 . 
   note that @xmath494 . 
   let @xmath495 and @xmath473 be such that @xmath496    by and , both @xmath497 and @xmath498 are supported in @xmath2 . using and we have @xmath499 therefore , @xmath500 since @xmath501 , the remainder @xmath502 satisfies . by hlder inequality , the right hand side of tends to @xmath392 , as @xmath503 . 
   this proves for @xmath484 .    as an immediate consequence of lemma [ lem012707 ] we obtain the following . 
   [ prop011007 ] if @xmath79 is lipschitz on @xmath504 , @xmath420 and @xmath505 , then either @xmath506 or @xmath507 .    by ( 
   * theorem 1.4.2 ( v ) ) , @xmath508 . 
   we apply @xmath79 to , and use lemma [ lem012707 ] . 
   in particular , we may consider truncations of @xmath20 at the level @xmath268 , @xmath509    [ cor - x1 ] if @xmath268 and @xmath510 , then @xmath511 . 
   we shall first prove the upper bound under the assumptions that @xmath6 is bounded and of zero divergence on the whole of @xmath49 , and @xmath512 . 
   [ prop021007 ] suppose that @xmath513 , @xmath495 , @xmath514 and @xmath20 is bounded . 
   then , @xmath515    we denote @xmath516 . by ( 
   * theorem 1.4.2 ( ii ) , ( iv ) ) , @xmath517 . 
   also , @xmath518 . considering , we observe that the right hand side of equals @xmath519 . by proposition [ prop011009 ] , 
   the left hand side of is @xmath520 however , the second term on the right hand side vanishes , because @xmath521dz=0,\ ] ] and @xmath522 , for a sufficiently large @xmath523 , is a first integral by virtue of corollary [ cor - x1 ] . 
   thus follows from . 
   the following elementary identity holds for functions @xmath524 , @xmath525 ^ 2+u^2(x)\frac{v(y)-v(x)}{v(x)}+u^2(y)\frac{v(x)-v(y)}{v(y)}\\ & & = v(x)v(y)\left[\frac{u(x)}{v(x)}-\frac{u(y)}{v(y)}\right]^2\nonumber.\end{aligned}\ ] ] let @xmath526 , @xmath527 , and @xmath528 . by 
   , @xmath529 ^ 2\ge \left\{\frac{w^2_n(x)}{\phi_a(x)+\eps}-\frac{w^2_n(y)}{\phi_a(y)+\eps}\right\}\left[\phi_a(x)-\phi_a(y)\right].\ ] ] multiplying both sides by @xmath530 and integrating over @xmath531 , we obtain @xmath532\nu(x - y)dxdy.\ ] ] using , we see that @xmath533 letting @xmath534 , we conclude that @xmath535 . letting @xmath536 and using ( * ? ? ? * part ( iii ) of theorem 1.4.2 ) , we obtain @xmath537 . 
   this proves . 
   [ sec5.2.3 ]    [ sec6 ]    suppose that @xmath538 , with @xmath474 , and @xmath539 on @xmath2 . here , as usual , @xmath2 is a bounded domain with the @xmath1 class boundary . by the discussion in section  [ sec : gener ] , @xmath540 has finitely many , say , @xmath541 connected components . 
   denote them by @xmath542 , and assume that @xmath543 belongs to the compactification of @xmath544 . 
   we start with the following extension result . 
   [ prop013110 ] there exist @xmath545 and @xmath546 such that @xmath547 @xmath548 has compact support and is smooth outside of @xmath549 .    according to the results of section 4 of @xcite we can find @xmath550 , supported in @xmath551 and such that @xmath552 for @xmath553 , numbers @xmath554 , and points @xmath555 such that @xmath556 here @xmath557 . 
   we denote by @xmath558 the right hand side of the above equality . 
   the field @xmath559 is incompressible on @xmath560 . 
   we choose @xmath545 so that @xmath561 for @xmath562 . 
   let @xmath563 equal to @xmath157 on @xmath564 and vanish outside of @xmath565 . 
   the field @xmath566 has the desired properties . 
   we consider the flow @xmath567 of measurable mappings generated by @xmath548 , which satisfies , and , see theorem iii.2 of @xcite . 
   condition needs to be modified as follows : there exists @xmath228 , such that @xmath568 let @xmath569 be a mollifier , let @xmath570 and let @xmath571 be the flow generated by @xmath572 . 
   it has been shown in section 3 of @xcite that @xmath573}\int_{b(0,n)}|\phi(x(t , x))-\phi(x_\eps(t , x))|\wedge 2^ndx=0\label{eq:2.31.10}\ ] ] for all @xmath574 and measurable functions @xmath575 . 
   we shall need the following modification of lemma [ lem012707 ] . 
   [ lem012707a ] if @xmath576 then , the conclusion of lemma [ lem012707 ] holds for the flow @xmath567 generated by @xmath548 . 
   suppose that @xmath577 . 
   since @xmath578 on @xmath2 , we can repeat the proof of the respective part of lemma [ lem012707 ] . 
   namely , keeping the notation from that lemma , for every @xmath481 we have @xmath579 such that latexmath:[\[\label{022807a } \int_{d }   w(x)u_0(x(-t , x))dx=\int_{d } w(x)u_0(x)dx,\quad    @xmath579 can be so adjusted that @xmath581 this part can not be guaranteed directly from the definition of the flow , as the extended field @xmath548 needs not be divergence - free . 
   equality holds however , when @xmath582 is replaced by @xmath583 for a sufficiently small @xmath495 . indeed , by the liouville theorem 
   , the jacobian @xmath584 of @xmath585 satisfies @xmath586 since @xmath587 in an open neighborhood of @xmath37 we conclude that @xmath588 on @xmath2 for ( sufficiently small ) @xmath589 . since @xmath20 and @xmath483 are supported in @xmath2 , @xmath590 letting @xmath503 and using we obtain . 
   the rest of the proof follows that of lemma [ lem012707 ] .    having established lemma [ lem012707a ] we proceed with the proofs of corollaries [ prop011007 ] and [ cor - x1 ] and proposition [ prop021007 ] with no alterations . 
   these results yield . 
   we consider the principal eigen - pair , say @xmath591 , of @xmath4 for the unit ball in @xmath46 . by rotation invariance of @xmath4 and uniqueness 
   , @xmath592 is a smooth radial function in the ball . 
   for @xmath553 we take radially symmetric functions @xmath593 , such that @xmath594 . 
   let @xmath595 , @xmath596 . 
   the vector field @xmath557 is of zero divergence and tangent to the spheres @xmath597 for all @xmath56 . indeed , since @xmath598 , @xmath599 as a result @xmath600 for any @xmath601 smooth radially symmetric function @xmath602 . 
   thus , we conclude that @xmath592 is the principal eigenfunction of @xmath44 for every @xmath24 . 
   we have @xmath603 , and @xmath604 attains its infimum , @xmath605 , at @xmath606 , see section  [ sec : lb ] . in passing we 
   note that the considered limiting eigenproblems are essentially different for different values of @xmath81 , in accordance with the fact that the `` escape rate '' @xmath607 of the isotropic @xmath81-stable lvy processes from @xmath2 depends on @xmath81 . 
   we refer the interested reader to @xcite , @xcite , @xcite for more information on the eigenproblem of @xmath4 , see also @xcite .      to complete the proof of theorem  [ main - thm ] we only need to explain the attainability of infimum appearing on the right hand side of . 
   this is done in the following . 
   [ lem1 ] if @xmath608 , then @xmath609 attains its infimum on the set . 
   if @xmath610 is the infimum , then we can choose functions @xmath611 in the set given in the statement of the lemma , such that @xmath612 and , by choosing a subsequence , that @xmath611 weakly converge to @xmath613 . 
   since @xmath614 is precompact in @xmath47 , we may further assume that @xmath615 in @xmath47 and a.e . 
   this implies that @xmath616 , @xmath617 is a first integral and , by and fatou s lemma , that @xmath618 . from the definition of @xmath619 
   , we conclude that equality actually occurs . 
   krzysztof bogdan thanks the department of mathematics at stanford university for hospitality during his work on the paper , lenya ryzhik for discussions on incompressible flows and michael frazier for a discussion on schur s test .
summary: we study the principal dirichlet eigenvalue of the operator @xmath0 , on a bounded @xmath1 regular domain @xmath2 . here 
   @xmath3 , @xmath4 is the fractional laplacian , @xmath5 , and @xmath6 is a bounded @xmath7-dimensional divergence - free vector field in the sobolev space @xmath8 . 
   we prove that the eigenvalue remains bounded , as @xmath9 , if and only if @xmath6 has non - trivial first integrals in the domain of the quadratic form of @xmath4 for the dirichlet condition .