PROPERTIES OF SCALES OF KATO CLASSES, BESSEL POTENTIALS, MORREY SPACES, AND A WEAK HARNACK INEQUALITY FOR NON-NEGATIVE SOLUTIONS OF ELLIPTIC EQUATIONS

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1 Electronic Journal of Differential Equations, Vol. 7 7, No. 9, pp. 7. ISSN: URL: or PROPERTIES OF SCALES OF KATO CLASSES, BESSEL POTENTIALS, MORREY SPACES, AND A WEAK HARNACK INEQUALITY FOR NON-NEGATIVE SOLUTIONS OF ELLIPTIC EQUATIONS RENÉ ERLÍN CASTILLO, JULIO C. RAMOS-FERNÁNDEZ, EDIXON M. ROJAS Abstract. In this article, we study some basic properties of the scale of Kato classes related with the Bessel kernel, Lorentz spaces, and Morrey spaces. Also we characterize the weak Harnack inequality for non-negative solutions of elliptic equations in terms of the Bessel kernel and the Kato classes of order α.. Introduction In this article we prove a weak Harnack inequality for non-negative solutions of elliptic differential equations of divergence form with potentials from the Kato class of order α. Namely, given a bounded domain Ω in, we consider the Shrödinger operator n Lu + V u a ij x ux + V xux, x Ω, x i x j i,j where the matrix Ax a ij x is symmetric, bounded, measurable and positive uniformly in x, i.e., λ ξ Axξ, ξ Λ ξ, x Ω, ξ for some < λ Λ. Given V L loc Ω, a function u H Ω is a weak solution of Lu + V u if and only if A u, ϕ dx + V udx, for all ϕ H Ω. Ω Ω In this study, we use a class of potential more general than the one considered by Mohammed [5]. The study there is based heavily on the use and properties of the approximation of the Green function and the Green function of the corresponding operator. We substitute the approximate Green function by an approximate kernel of Bessel potentials denoted by G r α, and the Green function by the Kernel of the Bessel potentials. Also, we relate the Kato class of order α with the Bessel and Riesz potentials. Mathematics Subject Classification. 35B5, 35J, 35J5. Key words and phrases. Kato class; Bessel potential; Riesz potential; Lorentz space; weak Harnack inequality. c 7 Texas State University. Submitted December, 6. Published March 3, 7.

2 R. E. CASTILLO, J. C. RAMOS-FERNÁNDEZ, E. M. ROJAS EJDE-7/9 The Kato class K n on the n-dimensional space was introduced and studied by Aizenman and Simon [, 7]. The definition of K n is based on a condition considered by Kato [4]. Similar function classes were defined by Schechter [6] and Stummed []. We refer the reader to [, 3, 6, 7] for more information concerning to Kato class and its applications. We set φv, r sup x Bx,r V y dy, x y n where Bx, r {y : x y < r}. The Kato class K n consists of locally integrable functions V on such that lim φv, r. r Davies and Hinz [3] introduced the scale K n,α of the Kato class of order α. For α > we set V y ηv r sup dy. x Bx,r x y The Kato class of order α consists of locally integrable functions V on such that lim ηv r. r. Kato class of order α In this section, we gather definitions and notation that will be used later. By L loc,u we denote the space of functions V such that V y dy <. sup x Bx, Definition.. The distribution function D V of a measurable function V is given by D V λ m{x : V x > λ} where m denotes the Lebesgue measure on. Definition.. Let V be a measurable function in. The decreasing rearrangement of V is the function V defined on [, by V t inf{λ : D V λ t} t. Definition.3 Lorentz spaces. Let V be a measurable function, we say that V belongs to Ln/α, α > if t α n V tdt < and V belongs to L n, if sup t α n V t <. t> Definition.4 Morrey spaces. Let V L loc Rn, for q, we say that V belongs to L,n/q if sup V y dy V x r n/q L,n/q <. Bx,r The following definition is a slight variant of the scale K n,α Kato class.

3 EJDE-7/9 PROPERTIES OF SCALES OF KATO CLASSES 3 Definition.5. Let V L loc Rn, we say that V belongs to K n,α if V y ηv r sup dy <. x x y Bx,r Next, we study some properties of the class K n,α. Lemma.6. Kn,α L loc,u Rn. Proof. Let V K n,α and fix r >. Then there exits a positive constant C > such that ηv r C. It follows that V y sup V y dy sup dy, α >. x x y Therefore, x r Bx,r Bx,r sup V y dy < AC x Bx,r where A /r. Finally, let Bx, n k Bx k, r, then n V y dy sup V y dy. Thus, sup x Bx, Therefore, Kn,α L loc,u Rn. x R k n Bx k,r sup V y dy <. x Bx, Lemma.7. Ln/α, K n,α, α >. Proof. Let V Ln/α, α >, then t α n V tdt <. Since V χ Bx,ε V, we have V χ Bx,ε V t. Then t α n V χ Bx,ε tdt Thus, V χ Bx,ε Ln/α,. On the other hand, letting gx x α n, we have t α n V tdt. m{x : gx > λ} m{x : x α n > λ} {x m : x < λ } n C n, λ where C n mb,. Next, we set t C n λ n, then λ C n n t α n. Thus g t C n t α n, from this we obtain g n, n, sup C n n t α α n t n C n t> n.

4 4 R. E. CASTILLO, J. C. RAMOS-FERNÁNDEZ, E. M. ROJAS EJDE-7/9 Finally, Bx,ε V y x y dt V χ Bx,ε n α, n Thus, V K n,α. n C, n V χ Bx,ε n α,. Example.8. Regarding the functions that belong to K n,α, we claim that V x x α log x α K n,α. Proof of the Claim. It will be sufficient to show that V Ln/α,, to do this let us consider { } { m x : x α log x α > λ m x : x α log x α < }. λ Putting ϕ x x α log x α, we have {x m : x α log x α < } {x } m : ϕ x < λ λ {x m : x < ϕ λ } C n ϕ n. λ Let t C n ϕ λ n, thus Cn /n ϕ λ t/n, then ϕ λ Cnt/n, where Cn Cn /n so λ ϕcnt/n, hence Therefore Note taht λ V t t α n V tdt Cn ϕ Cnt /n Cn t α n log t. α Cn t α n log t Cn α t α n log t. α t α n then V Ln/α,, hence V K n,α. dt t α n log t Cn α Lemma.9. If V L,n/q and p > n/α, p, then V y x y dy α n/p V L,n/q R. n Bx, Moreover L,n/q K n,α. Proof. Let V L,n/q. Note that V y V yχ Bx,y dy x y x y dy Bx, where dµy V y χ Bx, ydy, from this we have µbx, r V y χ Bx, ydy. Bx,r dt <, tlog t α dµy,. x y

5 EJDE-7/9 PROPERTIES OF SCALES OF KATO CLASSES 5 Going back to., we obtain dµy n α x y n α n α + n α n α Therefore L,n/q K n,α. r α n µbx, rdr r α n r α n + n α n/q Bx,r Bx, Bx,r r α n V y dydr Bx, r α n+ n q r n/q V y dydr V y dydr Bx,r r α n n/q V y dy dr Bx,r V y dy dr [ ] n α r α n p dr + n/q r α n dr V L,n/q [[ rα n/p n/q rα n n α α n + p α n [ pn α + αp n n α α n/pn α [ np n ] α n/p V pα n L,n/q R. n ] V L,n/q ] α n p V L,n/q 3. Space of functions of bounded mean oscillation BMO In the same sense that the Hardy space H is a substitute for L, it will turn out that the space BMO the space of bounded mean oscillation is the corresponding natural substitute for the space L of bounded functions on. A locally integrable function f belongs to BMO if fx f Br dm A 3. m holds for all balls Bx, r, here f Br m fdm fdm denotes the mean value of f over the ball and m stand for the Lebesgue measure on. The inequality 3. asserts that over any ball B, the average oscillation of f is bounded. The smallest bound A for which 3. is satisfied is then taken to be the norm of f in this space, and is denoted by f BMO. Let us begin by making some remarks about functions that are in BMO. The following result is due to Jhon-Niremberg. If f BM O then there exist positive constants C and C so that, for every r > and every ball m{x : fx f Br > λ} Ce Cλ/ f BMO m.

6 6 R. E. CASTILLO, J. C. RAMOS-FERNÁNDEZ, E. M. ROJAS EJDE-7/9 One consequence of the above result is the following corollary. Corollary 3.. If f BMO, then there exist positive constants C and C such that e C fx f Br C C dm C C + m for every ball and < C < C. Proof. Let us define ϕx e x. Notice that ϕ, and hence e C fx f Br C e Cλ m{x : fx f Br > λ}dλ [ ] CC e C Cλ dλ m. From the above inequality we have e C fx f Br dm CC C C + m. 4. p bounded mean oscillation A locally integrable function f belongs to BMO p if for p < /p f BMOp sup fx f Br p dm <. m Theorem 4.. If f BMO p then there exists a positive constant C depending on p such that f BMO C p f BMOp. Proof. Let f BMO p by virtue of the Hölder inequality we have fx f Br dm [m ] /p fx f Br p dm Hence fx f Br dm sup fx f br p dm m m for any ball. 5. Bessel kernel /p. The connection between the Bessel and Riesz potential was observed by Stein [8, 9]. We will develop the basic properties of the Bessel kernel. Here F : S S denotes the Fourier transform on S where S represent the set of all tempered distributions. S is thus the dual of the Schwartz space S. For f L we have F fξ fxe πix ξ dx. The Riesz kernel, I α, < α < n, is defined by /p I α x x α n γα, 5.

7 EJDE-7/9 PROPERTIES OF SCALES OF KATO CLASSES 7 where γα πn/ α Γα/ Γ n α/ Γ denotes the gamma function. We begin by deriving the kernel of the Bessel potential. First let us consider t a Γa e ts a d. 5. After a suitable change of variables is not difficult to obtain 5.. Using 5. with α/ > we have 4π α/ + 4π x α/ Γα/ e 4π +4π x α/ d. 5.3 Now we want to compute F { + 4π x α/ }ξ + 4π x α/ e πix ξ dx. By 5.3 we obtain e 4π α/ 4π +4π x α/ d e πix ξ dx R Γα/ n e π ξ 4π α/ e α n d 4π Γα/ ; therefore F { + 4π x α/ }ξ 5.. Bessel kernel. We define the Bessel kernel G α x e π x e α n 4π Lemma 5.. a For each α >, G α x L. b F G α x + 4π x α/. Proof. a By 5.4 we obtain G α xdx R 4π n α/ Γα/ Since e π x dx n/ and using Fubini, we set G α xdx e α n R 4π n α/ 4π Γα/ After a suitable change of variable we have G α xdx, e π ξ e α n d 4π. e π x e α n 4π e π x d. 5.4 d dx. d dx /. and so G α x L. b In the sense of distributions we have whenever ϕ S, fxf ϕxdx F fxϕxdx. 5.5

8 8 R. E. CASTILLO, J. C. RAMOS-FERNÁNDEZ, E. M. ROJAS EJDE-7/9 Let us consider the function By 5.5 we have fx e 4π e π x ; e 4π +4π x ϕxdx where ϕx F ϕx, then By Fubini s theorem, R 4π n α/ Γα/ then F fx e 4π e π x n/. e 4π e π x e 4π +4π x ϕxdx e 4π e π x e 4π +4π x e 4π e π x n/ ϕxdx, α/ d n/ ϕxdx α/ d α n d α/ d ϕxdx ϕxdx. That is, + 4π x α/ ϕxdx G α xϕxdx; therefore F G α x + 4π x α/. Remark 5.. From Lemma 5.b we have G α G β G α+β. Lemma 5.3. F { e π x a d } e Proof. By definition { F Therefore { F e π x a d } π x e π x a d } n/ a d. e π x a d e πix ξ dx e π x e πix ξ dx a d e π x / n/ a d. e π x / n/ a d. Proposition 5.4. x α n γα e π x / α n/ d. Proof. We have x α n e π x / α n/ d e π/u x u α n x du e π/u u α n du.

9 EJDE-7/9 PROPERTIES OF SCALES OF KATO CLASSES 9 therefore x α n πα n/ x α n Γ n α α π n/ Γα/ x α n x α n γα ; x α n γα e w π w α n π w dw e w w dw e π x / α n/ d. Remark 5.5. By 5. and Proposition 5.4 we can define I α x as follows I α x Γα/ 4π α/ e π x / α n/ d. 5.6 Comparing the formulas 5.4 and 5.6 it follows immediately that G α x is positive, and < G α x < I α x for < α < n. Proposition 5.6. G α x x α n γα + O x α n, as x. Proof. For ε > we have ε e π x / α n/ d e π/u x u α n x du ε x x α n x α n e π/u u α n ε x x π ε x π x α n π α n/ ε du e w π w α n π w dw e w w α n+ dw. Let us define ϕx, ε x π ε e w w α n+ dw, note that ϕx, ε as x. So we can write e π x / α n/ d C x α n ϕx, ε, ε where C π α n/. Now we have to prove that for every τ > there exists λ > such that if x < λ then Gα x x α n τ x α n. γα To do that let us consider G α x x α n γα e π x / [e /4π ] α n d,

10 R. E. CASTILLO, J. C. RAMOS-FERNÁNDEZ, E. M. ROJAS EJDE-7/9 since e/4π as, we have e /4π + Oe /4π as. e /4π Taking τ > there exists ε > such that therefore γατ x α n γα e π x / [e /4π ] α n τ x α n ; e π x / τ e /4π α n e π x / [e /4π ] α n d d d τ x α n. 5.7 Since ε > has been chosen we take x < λ such that ϕx, ε τ c. Then we obtain e π x / [e /4π ] α n d 4π α/ Γα/ ε e π x / τ ε e /4π α n d τ e π x / α n d 4π α/ Γα/ ε τ C x α n ϕx, ε τ x α n. Finally G α x x α n γα [ ε + from 5.6 and 5.7 we obtain therefore G α x x α n γα ε e π x / [e /4π ] α n G α x x α n τ x α n ; γα e π x / [e /4π ] α n e π x / [e /4π ] α n d ] O x α n as x for < α < n. On the other hand by differentiating formula 5.4 we obtain G αx C e π x e α n d 4π x j x j C x j e π x α n d d d

11 EJDE-7/9 PROPERTIES OF SCALES OF KATO CLASSES by Proposition 5.4, the above expression is less than or equal to C x j x α n. Thus G αx C x α n. x j Proposition 5.7. G α x Oe x as x, which shows that the kernel G α is rapidly decreasing as x. Proof. Let f e π x 4π. After a not too difficult calculation we obtain fπ x e x, which is a maximum value. Also if x then clearly Now let us consider e π x / e /4π e π/ e /4π, e π x / e /4π e x mine x, e π/ /4π { e x e π/ 4π for π x. Note π + 4π since a + b ab; therefore we have Finally when x, From this we obtain Therefore, G α x so G α x Me x, where x x π 8π. mine x, e π/ /4π e x M if x π + 4π, if x π + 4π. π e π. e π x / e /4π e x e π e 8π. e x e π e α n d 8π e π e α n d 8π. Remark 5.8. From Proposition 5.6, if < α < n then there exist C α > and C α > such that C α x α n G α x C α x α n for all x with < x <. Also from Proposition 5.7 we observe that, for every α > there exist M α > such that G α x M α e C x for all x with x >.

12 R. E. CASTILLO, J. C. RAMOS-FERNÁNDEZ, E. M. ROJAS EJDE-7/9 From these two observations we can write G α x C α χ B,x x + e C x for all x. Next we use the Bessel kernel to build a explicit weak solution for the Schrödinger operator. Let ϕ be a function belonging to C and such that { if x ϕx if x with ϕx for every x, we set ϕ r x ϕ x r and define for x r. Observe that G r αx G α xϕ r x G r αx G α x as r and that G r α H Ω L Ω with A G r α, ϕ Ω V G r αϕdm 5.8 for any ϕ H Ω and V L loc Ω. Gr α will be called an approximate Bessel kernel. 5.8 tell us that G r α is a weak solution of LG r α + V G r α. Also, for a real function f we write f + max{f, } for x Ω, and r > with Bx, r. 6. Main result In this section we give a characterization of the weak Harnack inequality for nonnegative solutions of elliptic equations in terms of the Bessel kernel and Kato class of order α. We start this section with the following result. Lemma 6.. Let u be a non-negative weak solution of Lu+V u. If φv r < for some r <, then there exists a constant C > such that for B r Ω with < r r. log u log u dm C Proof. Let ϕ C B r with ϕ on. Then A u ϕ dm u u Aϕ u u dm + Thus, u u Aϕ dm u A u ϕ dm + u a ϕ u ϕdm. u A ϕ u ϕdm u

13 EJDE-7/9 PROPERTIES OF SCALES OF KATO CLASSES 3 λ ϕ log u dm λ ϕ u u dm u u Aϕ dm u u u Aϕ u dm V u ϕ u dm + A ϕ u u ϕdm V ϕ dm + A ϕ u u ϕdm. Since V K n,α there exists C > such that ηv r C. Now, it follows that V y r V y dy dy. x y Thus, Bx,r This immediately gives us Bx,r Bx,r V y dy ηv rr Cr. log u dm Cr n. Bx,r From this and the Poincaré inequality we obtain log u log u dm C log u dm Cr. The above lemma and Theorem 4. tell us that log u BMO. Then by Corollary 3. there exits a positive constant C such that for β >, e fx f Br dm C, m where f log u. Using this we conclude that e βf dm e βf dm m m m e βf f Br dm e βf f Br dm e β f f Br dm C, m which implies hence e βf dm e βf dm C[m ], u β dm u β dm C[m ]. 6.

14 4 R. E. CASTILLO, J. C. RAMOS-FERNÁNDEZ, E. M. ROJAS EJDE-7/9 Proposition 6.. Suppose that 6. holds, then there exits a positive constant C such that u β dm C B r u β dm, where B r Ω. The above inequality is known as doubling condition. Proof. If 6. holds, then we have u β dm / / u β dm C / m ; from this inequality we obtain / u β dm C / m /. u β dm 6. On the other hand, by Schwartz s inequality and 6. we have m u β/ u β/ dm / / u β dm u β dm / / u β dm u β dm B r / /. C / m u β dm u β dm B r Thus B m C / m r u β dm. B r u β dm Finally u β dm C u β dm. B r We need the following mean-value inequality see, []. Theorem 6.3. Let u be a weak solution of Lu + V u in Ω. Given < p <, there are positive constants and C such that /p sup u C u p dm whenever φv r. Let J : R R be a smooth function. In our next result, we consider a weak solution of Lu+V Ju in Ω such that Ju u in Ω. The proof of the following theorem follows along the same lines as the corresponding proof on [5]. Theorem 6.4 Weak Harnack Inequality. Let u be a non-negative weak solution of Lu + V u, and let Bx, r such that 4 Ω. Then there are positive constants and C such that /β u β dm C inf fu, B r where β is the constant in 6., whenever ηv r.

15 EJDE-7/9 PROPERTIES OF SCALES OF KATO CLASSES 5 Proof. For t >, we write Ω r t {x Ω : G r αx > t} and Ω t {x Ω : G α x > t}, and also define the function G r Hr, t α log + G r α. t t On the one hand, we have log x/ x log x for x. On the other hand, we have [ ] log + Gr α t G r Hr, t α t and that Hr, t is supported on Ω r t for all t >. Now, we claim that given β > there is a positive constant C Cβ, λ, L such that for any t > u β/ log + Gr α t C [ ] dm V G r t αu β dm + u β dm. 6.3 Ω r t We first prove the claim for a solution of Lu + V Ju such that Ju u and inf Ω u >. In the definition 5.8 we take ϕ t +u β as a test function taking into account that inf Ω u >. Then, we find that A G r α, G r u α α G r α dm + α A G r α, u t +u β dm Using Ω r t t G r + u β dm. α Hr, tu β + βu β Hr, t u u β t G r + G r α α in 6.4 follows by application of 6. that A G r α, G r u β α Ω r G r t α dm + α A u β/, u β/ [ log + G r ] α dm Ω t u β t dm β A u, Hr, tu β dm from this we have u β/ log + Gr α t [ dm Cβ, λ Ω r t Ω Ω Ω r t G r α G r α 6.4 u β ] t dm β V JuHr, tu β dm. Ω Recalling that Ju u and using 6. we have 6.3. Now, let u be a nonnegative weak solution of Lu + V u in Ω. Then for any ε >, and Ju u ε, we can see that w u + ε is a weak solution of Lw + V Jw in Ω with Jw w such that inf Ω w >. Therefore using 6.4 for w, letting ε and using the fact that u is locally bounded, we should apply the Fatou s Lemma and the Lebesgue dominated convergence theorem to have the full statement of the claim. Let R j Cj t / for j, and C and C the constants in Remark 5.8. Then the following inclusions are direct consequences of the inequalities in Remark 5.8 B R Ω r t, Ω t t B R.

16 6 R. E. CASTILLO, J. C. RAMOS-FERNÁNDEZ, E. M. ROJAS EJDE-7/9 Since u, G r α belong to L loc Ω, we shall apply the Sobolev inequality to 6.4 to obtain the following chain of inequalities: C R Ω r t log + Gr α t u β dm C R C t Ω r t Ω r t log + Gr α t u β dm u β/ log + Gr α Ω r t t dm V G r αu β dm + m u β dm. Next, using Remark 5.8 and 6.4 in the last inequality, we obtain u β dm C t sup u β V x y α n dy + C u β dm. B R B R m R Ω r t Since G r αx Gx as r, we observe that χ Ωt lim inf r χ Ωt r from this and the Fatou s Lemma we deduce u β dm C [ ηv R sup u β + u β x ] Ω r t B t R by 6.4 we obtain R R B R u β dm C t [ ηv R sup B R u β + u β x ] and thus, let r > such that B 4r Ω. We choose t such that t max{c, C }r α n and observe that 6.4 holds if C C then R r and R r. If C < C, then R C C r and R r. In either case, we use the doubling property of u α and Theorem 6.3 to conclude that u β t dm ηv u β t dm + Cuβ, by choosing r sufficiently small, we conclude that which gives the desired result. u β Cu β x, References [] M. Aizenman, B. Simon; Browmian motion and Harnack s inequality for Schödinger operator, Comm. Pure Appl. Math., 35, 98 [] F. Chiarenza, E. Fabes, N. Garofals; Harnack s inequality for Schödinger operator and continuity of solutions, Proc. Amer. Math. Soc., 98, 986, [3] E.B. Davis, A. Hinz; Kato class potentials for higher order elliptic operators, J. London Math. Soc., , [4] T. Kato; Schödinger operator with singular potentials, Israel J. Math., 3, 973, [5] A. Mohammed; Weak Harnack s inequality for non-negative solutions of elliptic equations with potential, Proc. Amer. Math. Soc., 9,, [6] M. Schechter; Spectral of Partial Differential Operators, North-Holland, 986. [7] B. Simon; Schödinger semigroups, Bull. Amer. Math. Soc., 7, 98, [8] E. M. Stein; Harmonic Analysis Real-Variable Methods, orthogonality and oscillatory integral, Princeton University Press, Princeton NJ, 993. [9] E. M. Stein; Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton NJ, 97.

17 EJDE-7/9 PROPERTIES OF SCALES OF KATO CLASSES 7 [] F. Stummed; Singulare elliptische differential operator in Hilbertschen Räumen, Math. Ann., 3, 956, René Erlín Castillo Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia address: recastillo@unal.edu.co Julio C. Ramos-Fernández Departamento de Matemáticas, Universidad de Oriente, 6 Cumaán Edo. Sucre, Venezuela address: jcramos@udo.edu.ve Edixon M. Rojas Departamento de Matemáticas, Universidad Nacional de Colombia, Bogotá, Colombia address: emrojass@unal.edu.co