Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and Wk,p(·)

We study the Riesz potentials I_αf on the generalized Lebesgue spaces L^p(·)(?^d), where 0 < α < d and I_αf(x) ? ∫equation/tex2gif-inf-3.gif |f(y)| |x – y|^{α – d} dy. Under the assumptions that p locally satisfies |p(x) – p(x)| ≤ C/(– ln |x – y|) and is constant outside some large ball, we prove that I_α : L^p(·)(?^d) → L^p?(·)(?^d), where . If p is given only on a bounded domain Ω with Lipschitz boundary we show how to extend p to on ?^d such that there exists a bounded linear extension operator ? : W^1,p(·)(Ω) ? (?^d), while the bounds and the continuity condition of p are preserved. As an application of Riesz potentials we prove the optimal Sobolev embeddings W^k,p(·)(?^d) ?L^p*(·)(R^d) with and W^1,p(·)(Ω) ? L^p*(·)(Ω) for k = 1. We show compactness of the embeddings W^1,p(·)(Ω) ? L^q(·)(Ω), whenever q(x) ≤ p*(x) – ε for some ε > 0. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the Robin Boundary Condition for Laplace's Equation in Lipschitz Domains

Abstract

Let Ω be a bounded Lipschitz domain in ? ⁿ , n ≥ 3 with connected boundary. We study the Robin boundary condition ?u/?N + bu = f ∈ L ^p (?Ω) on ?Ω for Laplace's equation Δu = 0 in Ω, where b is a non-negative function on ?Ω. For 1 < p < 2 + ?, under suitable compatibility conditions on b, we obtain existence and uniqueness results with non-tangential maximal function estimate ‖(?u)*‖ _p  ≤ C‖f‖ _p , as well as a pointwise estimate for the associated Robin function. Moreover, the solution u is represented by a single layer potential.     

On an inequality by N. Trudinger and J. Moser and related elliptic equations

It has been shown by Trudinger and Moser that for normalized functions u of the Sobolev space ??^{1, N} (Ω), where Ω is a bounded domain in ?^N, one has ∫_Ω exp(α_N|u|^{N/(N ? 1)})dx ≤ C_N, where α_N is an explicit constant depending only on N, and C_N is a constant depending only on N and Ω. Carleson and Chang proved that there exists a corresponding extremal function in the case that Ω is the unit ball in ?^N. In this paper we give a new proof, a generalization, and a new interpretation of this result. In particular, we give an explicit sequence that is maximizing for the above integral among all normalized “concentrating sequences.” As an application, the existence of a nontrivial solution for a related elliptic equation with “Trudinger‐Moser” growth is proved. © 2002 John Wiley & Sons, Inc.     

Functions of normal operators under perturbations

In Peller (1980) [27], Peller (1985) [28], Aleksandrov and Peller (2009) [2], Aleksandrov and Peller (2010) [3], and Aleksandrov and Peller (2010) [4] sharp estimates for f(A)−f(B) were obtained for self-adjoint operators A and B and for various classes of functions f on the real line R. In this paper we extend those results to the case of functions of normal operators. We show that if a function f belongs to the Hölder class Λα(R²), 0<α<1, of functions of two variables, and N₁ and N₂ are normal operators, then ‖f(N₁)−f(N₂)‖?const‖fΛα_‖‖N₁−N₂α^‖. We obtain a more general result for functions in the space for an arbitrary modulus of continuity ω. We prove that if f belongs to the Besov class , then it is operator Lipschitz, i.e., . We also study properties of f(N₁)−f(N₂) in the case when f∈Λα(R²) and N₁−N₂ belongs to the Schatten–von Neumann class Sp.     

Modified Dyadic Integral and Fractional Derivative on ℝ+

For functions from the Lebesgue space L(?₊), we introduce the modified strong dyadic integral J _α and the fractional derivative D ^(α) of order α > 0. We establish criteria for their existence for a given function f ∈ L(?₊). We find a countable set of eigenfunctions of the operators D ^(α) and J _α, α > 0. We also prove the relations D ^(α)(J _α(f)) = f and J _α(D ^(α)(f)) = f under the condition that $\smallint _{\mathbb{R}_ + } f(x)dx = 0$ . We show the unboundedness of the linear operator $J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )$ , where L _{J α} is its natural domain of definition. A similar assertion is proved for the operator $D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )$ . Moreover, for a function f ∈ L(?₊) and a given point x ∈ ?₊, we introduce the modified dyadic derivative d ^(α)(f)(x) and the modified dyadic integral j _α(f)(x). We prove the relations d ^(α)(J _α(f))(x) = f(x) and j _α(D ^(α)(f)) = f(x) at each dyadic Lebesgue point of the function f.     

Higher moments for kinetic equations: The Vlasov–Poisson and Fokker–Planck cases

Let us consider a solution f(x,v,t)?L¹(?^2N × [0,T]) of the kinetic equation where |v|^α+1 f_o,|v|^α ?L¹ (?2N × [0, T]) for some α< 0. We prove that f has a higher moment than what is expected. Namely, for any bounded set K_x, we have We use this result to improve the regularity of the local density ρ(x,t) = ∫?dν for the Vlasov–Poisson equation, which corresponds to g = E?, where E is the force field created by the repartition ? itself. We also apply this to the Bhatnagar-Gross-;Krook model with an external force, and we prove that the solution of the Fokker-Pianck equation with a source term in L² belongs to L²([0, T]; H^1/2(?)).     

Равносходимость разложений в кратный ряд и интеграл фурье, «прямоугольные частичные суммы» которых рассматриваются по некоторой подпоследовательности

In this paper we investigate the problem of the equiconvergence on T ^N = [-π, π) ^N of the expansions in multiple trigonometric series and Fourier integral of functions f ∈ L _p (T ^N ) and g ∈ L _p (? ^N ), where p > 1, N ≥ 3, g(x) = f(x) on T ^N , in the case when the “rectangular partial sums” of the indicated expansions, i.e.,– _n (x; f) and J _α(x; g), respectively, have indices n ∈ ? ^N and α ∈ ? ^N (n _j = [α _j ], j = 1,...,N, [t] is the integer part of t ∈ ?¹), in those certain components are the elements of “lacunary sequences”.     

Approximation by antiderivatives

Let Ω ?C be an open set with simply connected components and suppose that the functionφ is holomorphic on Ω. We prove the existence of a sequence {φ ^(?n)} ofn-fold antiderivatives (i.e., we haveφ ⁽⁰⁾(z)∶=φ(z) andφ ^(?n)(z)=dφ ^(?n?1)(z)/dz for alln ∈ N₀ and z ∈ Ω) such that the following properties hold:

1. For any compact setB ?Ω with connected complement and any functionf that is continuous onB and holomorphic in its interior, there exists a sequence {n _k} such that {φ^?nk} converges tof uniformly onB.
2. For any open setU ?Ω with simply connected components and any functionf that is holomorphic onU, there exists a sequence {m _k} such that {φ^?mk} converges tof compactly onU.
3. For any measurable setE ?Ω and any functionf that is measurable onE, there exists a sequence {p _k} such that {φ ^(-Pk)} converges tof almost everywhere onE.

     

An integral operator related to the Stokes system in exterior domains

Consider a bounded domain Ω in ?³ with C²-boundary ?Ω. In [1] the Stokes problem in the exterior domain ?³/Ω , with resolvent parameter [λ??\] ? [∞,0], is solved by using the method of integral equations. However, for estimating the corresponding solutions in L^p norms, it turns out that a certain operator defined on the spaces L^r(?Ω)³, for r ?]1, ∞[, has to be evaluated in the norm of L^r(?Ω)³. This estimate is proved in the present paper.     

Continuous Linear Extension of Ultradifferentiable Functions of Beurling Type

For a weight function ω and a closed set A ? ?^N let ?_(ω)(A) denote the space of all ω-Whitney jets of Beurling type on A. It is shown that for each closed set A ? ?^N there exists an ω-extension operator EA: ?_(ω)(A) → ?_(ω)(?^N) if and only if ω is a (DN)-function (see MEISE and TAYLOR [18], 3.3). Moreover for a fixed compact set K ? ?^N there exists an ω-extension operator E_K: ?_(ω)(K) → ?_(ω)(?^N) if and only if the Fréchet space ?_(ω)(K) satisfies the property (DN) (see Vogt [29], 1.1.).     

Schauder estimates for oblique derivative problems

We prove that the solution of the oblique derivative parabolic problem in a noncylindrical domain Ω^T belongs to the anisotropic Holder space C^{2+α, 1+α/2}(gwT) 0 < α < 1, even if the nonsmooth “lateral boundary” of Ω^T is only of class C^{1+α, (1+α)/2}). As a corollary, we also obtain an a priori estimate in the Hölder space C^2+α(Ω₀) for a solution of the oblique derivative elliptic problem in a domain Ω₀ whose boundary belongs only to the classe C^1+α.     

A direct theorem for strictly convex domains in ℂn

For a strictly covex C²-smooth domain Ω??ⁿ and a function f ? Λ^α(Ω) holomorphic in Ω, we construct polynomials p_N, deg p_N<-N, such that $$\left| {f(z) - p_N (z)} \right| \leqslant CN^{ - \alpha } ,z \in \bar \Omega .$$ Bibliography: 12 titles.     

A Remark on the Sobolev Regularity of Classical Solutions of Strongly Elliptic Equations

We prove that C^2,α(Ω ) solutions of problem (1.2) below are in H^m+2(Ω) for all m ∈ ?, if f and the coefficients are in H^m (Ω) n C^0,α (Ω ) Previously, this result was explicitly known only if m> n/2 (or if m = 0). A similar result holds for the quasi-linear equation (1.11) below.     

On an Approximate Solution of the Dirichlet Problem for the Generalized Laplacian

For an arbitrary differential operator P of order p on an open set X ? R ⁿ, the Laplacian is defined by Δ = P*P. It is an elliptic differential operator of order 2p provided the symbol mapping of P is injective. Let O be a relatively compact domain in X with smooth boundary, and B_j(j = 0…,p — 1) be a Dirichlet system of order p ? 1 on ?O. By {C_j} we denote the Dirichlet system on ?O adjoint for {B_j} with respect to the Green formula for P. The Hardy space H²(O) is defined to consist of all the solutions f of Δf = 0 in O of finite order of growth near the boundary such that the weak boundary values of the expression {B_jf} and {C_j(Pf)} belong to the Lebesgue space L²(?O). Then the Dirichlet problem consists of finding a solution f ? H²(O) with prescribed data {B_jf} on ?O. We develop the classical Fischer-Riesz equations method to derive a solvability condition of the Dirichlet problem as well as an approximate formula for solutions.     

Boundary blow-up solutions for a class of quasilinear elliptic equations

For a given bounded domain Ω in R ^N with smooth boundary ?Ω, we give sufficient conditions on f so that the m-Laplacian equation △ _m u = f(x, u, ?u) admits a boundary blow-up solution u ∈ W ^1,p (Ω). Our main results are new and extend the results in J.V. Concalves and Angelo Roncalli [Boundary blow-up solutions for a class of elliptic equations on a bounded domain, Appl. Math. Comput. 182 (2006), pp. 13–23]. Our approach employs the method of lower–upper solution theorem, fixed point theory and weak comparison principle.     

Zero density of L-functions related to Maass forms

Let f(z) be a Hecke-Maass cusp form for SL ₂(?), and let L(s, f) be the corresponding automorphic L-function associated to f. For sufficiently large T, let N(σ, T) be the number of zeros ρ = β +iγ of L(s, f) with |γ| ? T, β ? σ, the zeros being counted according to multiplicity. In this paper, we get that for 3/4 ? σ ? 1 ? ?, there exists a constant C = C(?) such that N(σ,T) ? T ^2(1?σ)/σ(logT) ^C , which improves the previous results.     

Solutions with boundary‐layers and spike‐layers to singularly perturbed quasilinear Dirichlet problems

The structure of nontrivial nonnegative solutions to singularly perturbed quasilinear Dirichlet problems of the form –?Δ_pu = f(u) in Ω, u = 0 on ?Ω, Ω ? R ^N a bounded smooth domain, is studied as ? → 0⁺, for a class of nonlinearities f(u) satisfying f(0) = f(z₁) = f(z₂) = 0 with 0 < z₁ < z₂, f < 0 in (0, z₁), f > 0 in (z₁, z₂) and f(u)/u^p–1 = –∞. It is shown that there are many nontrivial nonnegative solutions with spike‐layers. Moreover, the measure of each spike‐layer is estimated as ? → 0⁺. These results are applied to the study of the structure of positive solutions of the same problems with f changing sign many times in (0,∞). Uniqueness of a solution with a boundary‐layer and many positive intermediate solutions with spike‐layers are obtained for ? sufficiently small. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Zur Peano-Abbildung Einer Strecke

LetN ₁,N ₂ be two disjoint non-void subsets ofN={1, 2, ...}. It is shown that ifN ₂ is an infinite set, than there exists a Peano-mappingf of [0, 1] onto × ₁ ^∞ [0, 1] such thatf(t ₁+t ₂)=f(t ₂) holds, for arbitrary pointst ₁,t ₂ of the corresponding subsetsC[N ₁],C[N ₂] of the Cantor discontinuumC[N] of [0, 1].      

On the Structure of ∞-Harmonic Maps

Let H ∈ C ²(? ^N×n ), H ≥ 0. The PDE system arises as the Euler-Lagrange PDE of vectorial variational problems for the functional E _∞(u, Ω) = ‖H(Du)‖ _{L ^∞(Ω)} defined on maps u: Ω ? ? ⁿ  → ? ^N . (1) first appeared in the author's recent work. The scalar case though has a long history initiated by Aronsson. Herein we study the solutions of (1) with emphasis on the case of n = 2 ≤ N with H the Euclidean norm on ? ^N×n , which we call the “∞-Laplacian”. By establishing a rigidity theorem for rank-one maps of independent interest, we analyse a phenomenon of separation of the solutions to phases with qualitatively different behaviour. As a corollary, we extend to N ≥ 2 the Aronsson-Evans-Yu theorem regarding non existence of zeros of |Du| and prove a maximum principle. We further characterise all H for which (1) is elliptic and also study the initial value problem for the ODE system arising for n = 1 but with H(·, u, u′) depending on all the arguments.     

On the noise sensitivity of monotone functions

It is known that for all monotone functions f : {0, 1}ⁿ → {0, 1}, if x ∈ {0, 1}ⁿ is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability ? = n^?α, then P[f(x) ≠ f(y)] < cn^?α+1/2, for some c > 0. Previously, the best construction of monotone functions satisfying P[f_n(x) ≠ f_n(y)] ≥ δ, where 0 < δ < 1/2, required ? ≥ c(δ)n^?α, where α = 1 ? ln 2/ln 3 = 0.36907 …, and c(δ) > 0. We improve this result by achieving for every 0 < δ < 1/2, P[f_n(x) ≠ f_n(y)] ≥ δ, with:

• ? = c(δ)n^?α for any α < 1/2, using the recursive majority function with arity k = k(α);
• ? = c(δ)n^?1/2log^tn for t = log₂ = .3257 …, using an explicit recursive majority function with increasing arities; and
• ? = c(δ)n^?1/2, nonconstructively, following a probabilistic CNF construction due to Talagrand.

We also study the problem of achieving the best dependence on δ in the case that the noise rate ? is at least a small constant; the results we obtain are tight to within logarithmic factors. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 23: 333–350, 2003