On a reverse Hölder inequality for Schrödinger operators
aa r X i v : . [ m a t h . A P ] M a y ON A REVERSE H ¨OLDER INEQUALITY FOR SCHR ¨ODINGEROPERATORS
SEONGYEON KIM AND IHYEOK SEO
Abstract.
We obtain a reverse H¨older inequality for the eigenfuctions of theSchr¨odinger operator with slowly decaying potentials. The class of potentialsincludes singular potentials which decay like | x | − α with 0 < α <
2, especiallythe Coulomb potential. Introduction
In this paper we are concerned with a reverse H¨older inequality for the eigenfunc-tions of the Schr¨odinger operator − ∆ + V ( x ) in L ( R n ). More generally, we considersecond-order elliptic operators of the form Lu = − n X i,j =1 ∂∂x j ( a ij ( x ) u x i ) + V ( x ) u where a ij ( x ) is a measurable and real-valued function, and the matrix ( a ij ( x )) n × n isuniformly elliptic. Namely, there exists a positive constant Λ such that n X i,j =1 a ij ( x ) ξ i ξ j ≥ Λ | ξ | (1.1)for x, ξ ∈ R n . Particularly when a ij = δ ij (Kronecker delta function), the operator L becomes equivalent to the Schr¨odinger operator. In this regard, we shall call areal-valued function V ( x ) the potential.The reverse H¨older inequalities for solutions to the following Dirichlet boundaryproblem have been studied for a long time: Lu = − n X i,j =1 ∂∂x j ( a ij ( x ) u x i ) + V ( x ) u = λu in Ω u = 0 on ∂ Ω (1.2)where Ω is a bounded region in R n . When n = 2, Payne and Rayner [6] showed thatif λ is the first eigenvalue and u is the corresponding eigenfunction of the problem Mathematics Subject Classification.
Primary: 26D15; Secondary: 35J10.
Key words and phrases.
Reverse H¨older inequality, Schr¨odinger operator.This research was supported by the NRF grant funded by the Korea government(MSIP) (No.2017R1C1B5017496). (1.2) with a ij = δ ij and V ( x ) ≡ ( − ∆ u = λu in Ω u = 0 on ∂ Ω , then the following reverse Schwarz inequality holds: k u k L (Ω) ≤ r λ π k u k L (Ω) , which was extended to higher dimensions by Kohler-Jobin [5] (see also [7]). In thegeneral setting (1.2), the reverse H¨older inequalities, k u k L q (Ω) ≤ C p,q,λ,n k u k L p (Ω) , q ≥ p > , were obtained later by Talenti [8] for q = 2 and p = 1, and by Chiti [1] for all q ≥ p >
0, but with a nonnegative potential V ≥ a ij = a ji .Our aim in this paper is to remove these restrictions. Namely, we obtain a reverseH¨older inequality for solutions of (1.2) where V is allowed to be negative and we donot need to assume the symmetry, a ij = a ji .Before stating our results, we introduce the Morrey-Campanato class L α,r of po-tentials V , which is defined for α > ≤ r ≤ n/α by V ∈ L α,r ⇔ sup x ∈ R n ,ρ> ρ α − n/r Z B ρ ( x ) | V ( y ) | r dy ! /r < ∞ , where B ρ ( x ) is the ball centered at x with radius ρ . In particular, L α,n/α = L n/α and1 / | x | α ∈ L n/α, ∞ ⊂ L α,r if 1 ≤ r < n/α . Let us next make precise what we mean bya weak solution of the problem (1.2). We say that a function u ∈ H (Ω) is a weaksolution if Z Ω n X i,j =1 a ij ( x ) u x i φ x j dx + Z Ω V ( x ) u ( x ) φ ( x ) dx = Z Ω λu ( x ) φ ( x ) dx (1.3)for every φ ∈ H (Ω). Our result is then the following theorem. Theorem 1.1.
Let n ≥ . Assume that u ∈ H (Ω) is a weak solution of the problem (1.2) with λ ∈ R and V ∈ L α,r for α < and r > /α . Then we have k u k L q (Ω) ≤ CC n p α max { p, } np (2 − α ) (cid:16) nn − (cid:17) n ( n − p (2 − α ) k u k L p (Ω) (1.4) for all q ≥ p > . Here, C is a constant depending on Λ , λ, p, q, n and Ω , and C α = 1 + α α − α (cid:16) C n Λ k V k L α,r (cid:17) / (2 − α ) with a constant C n depending on n and arising from the Fefferman-Phong inequality (2.6) .Remark . The class L α,r , α <
2, of potentials in the theorem includes the positivehomogeneous potentials a | x | − α with a > < α < EVERSE H ¨OLDER INEQUALITY 3
The rest of this paper is organized as follows: In Section 2 we prove Theorem1.1. Compared with the previous results [5, 1] based on rearrangements of functions,our approach works also for negative potentials and for non-symmetric coefficients a ij . It is completely different approach and is based on a combination between theFefferman-Phong inequality and the classical Moser’s iteration technique.Throughout this paper, we denote A . B to mean A ≤ CB with unspecifiedconstant C >
Proof of Theorem 1.1
In this section we prove the reverse H¨older inequality (1.4). Since a complex-valued solution u satisfies (1.3) for every complex φ ∈ H (Ω), one can easily see thatreal and imaginary parts of the solution also satisfy (1.3) for every real φ ∈ H (Ω).On the other hand, once we prove the inequality for the real and imaginary parts, weget the same inequality for u . Indeed, using the inequality ( a + b ) s ≤ C ( a s + b s ) for a, b > s >
0, one can see k u k qL q (Ω) = Z Ω ( | Re u + i Im u | ) q/ dx = Z Ω ( | Re u | + | Im u | ) q/ dx ≤ C Z Ω | Re u | q + | Im u | q dx ≤ C ( k Re u k qL q (Ω) + k Im u k qL q (Ω) ) ≤ C ( k Re u k qL p (Ω) + k Im u k qL p (Ω) ) ≤ C k u k qL p (Ω) . Hence we may assume that the solution u is a real-valued function.Now we decompose u into two parts, f = max { u, } and g = − min { u, } . Thenit is enough to prove that (1.4) holds for f and g . Indeed, k u k L q (Ω) = k f − g k L q (Ω) ≤ k f k L q (Ω) + k g k L q (Ω) ≤ C ( k f k L p (Ω) + k g k L p (Ω) ) ≤ C k u k L p (Ω) . We only consider f because the proof for g follows obviously from the same argument.To prove (1.4) for f , we now divide cases into two parts, p ≥ p < The case p ≥ . In this case we will show that for all τ ≥ k f k L τω (Ω) . C /τα τ τ (2 − α ) k f k L τ (Ω) (2.1)with ω = n/ ( n − τ = p , we then iterate as τ = p, pω, pω , · · · , toobtain (1.4). Indeed, first put τ i = pω i SEONGYEON KIM AND IHYEOK SEO for i = 0 , , , · · · . Since τ i = τ i − ω , we then get for i = 1 , , · · · , k f k L τi (Ω) . C τi − α τ i − − α ) τi − k f k L τi − (Ω) = C pωi − α (cid:0) pω i − (cid:1) p (2 − α ) ωi − k f k L τi − (Ω) , which implies by iteration that k f k L τi (Ω) . (cid:0) C α p / (2 − α ) (cid:1) P ik =1 ( pω k − ) − (cid:0) ω / (2 − α ) (cid:1) P ik =1 ( k − pω k − ) − k f k L p (Ω) . Since ω = n/ ( n − >
1, by letting i → ∞ , this implies k f k L q (Ω) . k f k L ∞ (Ω) . C n p α p np (2 − α ) (cid:16) nn − (cid:17) n ( n − p (2 − α ) k f k L p (Ω) as desired.It remains to show (2.1). Since f ∈ H (Ω) is a positive part of the weak solution u , it follows that Z Ω n X i,j =1 a ij ( x ) f x i φ x j dx + Z Ω V ( x ) f ( x ) φ ( x ) dx = Z Ω λf ( x ) φ ( x ) dx (2.2)for every real φ ∈ H (Ω) supported on { x ∈ R n : u ( x ) > } . For l > m > f = f + l , and let ˜ f m = ( l + m if f ≥ m, ˜ f if f < m. We now consider the following test function φ = ˜ f βm ˜ f − l β +1 ∈ H (Ω)for β ≥
0. We then compute φ x j = β ˜ f β − m ( ˜ f m ) x j ˜ f + ˜ f βm ˜ f x j = β ˜ f βm ( ˜ f m ) x j + ˜ f βm ˜ f x j using the fact that( ˜ f m ) x j = 0 in { x : f ( x ) ≥ m } and ˜ f m = ˜ f in { x : f ( x ) < m } . (2.3)Substituting φ into (2.2) and using (2.3) together with the trivial fact f x i = ˜ f x i ,the first term on the left-hand side of (2.2) is written as Z Ω n X i,j =1 a ij ( x ) f x i φ x j dx = β Z Ω ˜ f βm n X i,j =1 a ij ( x )( ˜ f m ) x i ( ˜ f m ) x j dx + Z Ω ˜ f βm n X i,j =1 a ij ( x ) ˜ f x i ˜ f x j dx. Then it follows from the ellipticity (1.1) that Z Ω n X i,j =1 a ij ( x ) f x i φ x j dx ≥ Λ β Z Ω ˜ f βm |∇ ˜ f m | dx + Λ Z Ω ˜ f βm |∇ ˜ f | dx. (2.4) EVERSE H ¨OLDER INEQUALITY 5
Combining (2.2) and (2.4), we conclude thatΛ β Z Ω ˜ f βm |∇ ˜ f m | dx + Λ Z Ω ˜ f βm |∇ ˜ f | dx ≤ Z Ω − V f ( ˜ f βm ˜ f − l β +1 ) dx + Z Ω λf ( ˜ f βm ˜ f − l β +1 ) dx. Note here that |∇ ( ˜ f β/ m ˜ f ) | ≤ β ) (cid:16) β ˜ f βm |∇ ˜ f m | + ˜ f βm |∇ ˜ f | (cid:17) which follows by a direct computation together with (2.3). We therefore get Z Ω |∇ ( ˜ f β/ m ˜ f ) | dx ≤ β )Λ Z Ω − V f ( ˜ f βm ˜ f − l β +1 ) dx + 2 λ (1 + β )Λ Z Ω f ( ˜ f βm ˜ f − l β +1 ) dx ≤ β )Λ Z Ω | V | ˜ f βm ˜ f dx + 2 | λ | (1 + β )Λ Z Ω ˜ f βm ˜ f dx. (2.5)To control the term involving the potential in (2.5), we now use the so-calledFefferman-Phong inequality ([4]), Z R n | g | v ( x ) dx ≤ C n k v k L ,r Z R n |∇ g | dx, (2.6)where C n is a constant depending on n , and 1 < r ≤ n/
2. (It is not valid for r = 1as remarked in [2].) Applying this inequality along with H¨older’s inequality, the firstintegral on the right-hand side of (2.5) is bounded as Z Ω | V | ˜ f βm ˜ f dx ≤ (cid:18) Z Ω | V | α ˜ f βm ˜ f dx (cid:19) α (cid:18) Z Ω ˜ f βm ˜ f dx (cid:19) − α ≤ C n k| V | α k α L , ˜ r (cid:18) Z Ω |∇ ( ˜ f β/ m ˜ f ) | dx (cid:19) α (cid:18) Z Ω ˜ f βm ˜ f dx (cid:19) − α for all 1 < ˜ r ≤ n/
2. We note here that k| V | α k α L , ˜ r = k V k L α, r/α and apply Young’sinequality ab ≤ α εa ) /α + 2 − α ε − b ) / (2 − α ) with 0 < α < ε > Z Ω | V | ˜ f βm ˜ f dx ≤ C n k V k L α, r/α (cid:18) α ε /α Z Ω |∇ ( ˜ f β/ m ˜ f ) | dx + 2 − α ε − / (2 − α ) Z Ω ˜ f βm ˜ f dx (cid:19) . (2.7)By setting r = 2˜ r/α and taking ε /α = c (1 + β ) − with c = Λ2 αC n k V k L α,r so that C n k V k L α,r α ε /α β )Λ = 1 / , Since e r >
1, setting r = 2˜ r/α determines the condition r > /α in the theorem. SEONGYEON KIM AND IHYEOK SEO the gradient term in (2.7) can be absorbed into the left-hand side of (2.5), as follows: Z Ω |∇ ( ˜ f β/ m ˜ f ) | dx ≤ c − − α (cid:18) − αα (cid:19) ( β + 1) − α Z Ω ˜ f βm ˜ f dx + 2 | λ | (1 + β )Λ Z Ω ˜ f βm ˜ f dx. (2.8)Finally, applying the Gagliardo-Nirenberg-Sobolev inequality ([3]) to the left-handside of (2.8), we see (cid:18) Z Ω | ˜ f β/ m ˜ f | ω dx (cid:19) /ω . Z Ω |∇ ( ˜ f β/ m ˜ f ) | dx with ω = n/ ( n − f m ≤ ˜ f and setting β + 2 = τ , we thereforeget (cid:18) Z Ω ˜ f τωm dx (cid:19) /ω ≤ c − − α (cid:18) − αα (cid:19) ( τ − − α Z Ω ˜ f τ dx + 2 | λ | ( τ − Z Ω ˜ f τ dx. which implies the desired estimate (cid:18) Z Ω f τω dx (cid:19) /ω . (cid:16) α α − α (cid:16) C n Λ k V k L α,r (cid:17) − α (cid:17) τ − α Z Ω f τ dx. by letting m → ∞ and l → The case p < . From the case p = 2, we have k f k L ∞ (Ω) < ∞ and k f k L ∞ (Ω) . C n α n − α ) (cid:16) nn − (cid:17) n ( n − − α ) k f k L (Ω) ≤ C n α n − α ) (cid:16) nn − (cid:17) n ( n − − α ) k f k (2 − p ) / L ∞ (Ω) k f k p/ L p (Ω) ≤ k f k L ∞ (Ω) + C n p α p (cid:16) − p (cid:17) − p np (2 − α ) (cid:16) nn − (cid:17) n ( n − p (2 − α ) k f k L p (Ω) . (2.9)For the third inequality, we used here Young’s inequality, ab ≤ (cid:16) − p (cid:17) ( ǫa ) − p + p ǫ − b ) p with ǫ = ( − p ) (2 − p ) / , a = k f k (2 − p ) / L ∞ (Ω) and b = C n α n − α ) (cid:16) nn − (cid:17) n ( n − − α ) k f k p/ L p (Ω) . By absorbing the first term on the right-hand side of (2.9) into the left-hand side, weconclude that k u k L q (Ω) . k u k L ∞ (Ω) . C n p α np (2 − α ) (cid:16) nn − (cid:17) n ( n − p (2 − α ) k u k L p (Ω) as desired. EVERSE H ¨OLDER INEQUALITY 7
References [1] G. Chiti,
A reverse H¨older inequality for the eigenfunctions of linear second order elliptic oper-ators , Z. Angew. Math. Phys. 33 (1982), 143-148.[2] D. Danielli,
A Fefferman-Phong type inequality and applications to quasilinear subelliptic equa-tions , Potential Anal. 11 (1999), 387-413.[3] L. C. Evans,
Partial differential equations , Graduate Studies in Mathematics, 19. AmericanMathematical Society, Providence, RI, 1998.[4] C. Fefferman,
The uncertainty principle , Bull. Amer. Math. Soc. 9 (1983), 129-206.[5] M. T. Kohler-Jobin,
Sur la premi`ere fonction propre d’une membrane: une extension `a N di-mensions de l’in´egalit´e isop´erim´etrique de Payne-Rayner , Z. Angew. Math. Phys. 28 (1977),1137-1140.[6] L. E. Payne and M. E. Rayner,
An isoperimetric inequality for the first eigenfunction in the fixedmembrane problem , Z. Angew. Math. Phys. 23 (1972), 13-15.[7] L. E. Payne and M. E. Rayner,
Some isoperimetric norm bounds for solutions of the Helmholtzequation , Z. Angew. Math. Phys. 24 (1973), 105-110.[8] G. Talenti,
Elliptic equations and rearrangements , Ann. Scuola Norm. Sup. Pisa (4) 3 (1976),697-718.
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Ko-rea
E-mail address : [email protected] E-mail address ::