Mixed Lebesgue Spaces Research Articles

A note on weak existence for singular SDEs

Recently Krylov [N. V. Krylov, On time inhomogeneous stochastic Itô equations with drift in [Formula: see text], Ukraïn. Mat. Zh. 72 (2020) 1232–1253] established weak existence of solutions to SDEs for integrable drifts in mixed Lebesgue spaces, whose exponents satisfy the condition [Formula: see text], thus going below the celebrated Ladyzhenskaya–Prodi–Serrin condition. We present here a variant of such result, whose proof relies on an alternative technique, based on a partial Zvonkin transform; this allows for drifts with growth at infinity and/or in uniformly local Lebesgue spaces.

Robustness of average sampling and reconstruction of shift-invariant signals in mixed Lebesgue spaces

Robustness of average sampling and reconstruction of shift-invariant signals in mixed Lebesgue spaces

Uniform convergence of the one-dimensional cubic Schrödinger equation

In this article, we investigate the uniform convergence problem of the one-dimensional cubic Schrödinger equation. By using the Kato smoothing estimate, the maximal function estimate and the dyadic mixed Lebesgue spaces, we establish the uniform convergence of the one-dimensional cubic Schrödinger equation in H s ( R ) ( s > 1 6 ) which is an alternative proof of Theorem 1.1 ( n = 1 , p = 3 ) of Compaan et al. [Pointwise convergence of the Schrödinger flow. Int Math Res Not. 2021;1:596–647].

Regularity criteria of the axisymmetric Navier-Stokes equations and Hardy-Sobolev inequality in mixed Lorentz spaces

In this paper, we refine some known regularity criteria of the 3D axisymmetric Navier-Stokes equations via establishing several new Hardy-Sobolev inequalities in mixed Lebesgue spaces and mixed Lorentz spaces. These inequalities cover many known corresponding results and are of independent interest.

Sampling and Reconstruction of Concentrated Reproducing Kernel Signals in Mixed Lebesgue Spaces

Sampling and Reconstruction of Concentrated Reproducing Kernel Signals in Mixed Lebesgue Spaces

On approximation of double Fourier series and its conjugate series for functions in mixed Lebesgue space

In this paper, we study the approximation of double Fourier series and its conjugate series for functions in mixed Lebesgue space Lp, p ∈ [1,∞]2 using double Karamata Kλ,μ means.

Reconstruction and Error Analysis Based on Multiple Sampling Functionals over Mixed Lebesgue Spaces

Reconstruction and Error Analysis Based on Multiple Sampling Functionals over Mixed Lebesgue Spaces

Random sampling and reconstruction in reproducing kernel subspace of mixed Lebesgue spaces

In this article, we consider the random sampling in the image space of an idempotent integral operator on mixed Lebesgue space . We assume some decay and regularity conditions on the integral kernel and show that the bounded functions in can be approximated by an element in a finite‐dimensional subspace of on . Consequently, we show that the set of bounded functions concentrated on is totally bounded and prove with an overwhelming probability that the random sample set uniformly distributed over is a stable set of sampling for the set of concentrated functions on . Further, we propose an iterative scheme to reconstruct the concentrated functions from their random measurements.

Random Average Sampling and Reconstruction in Shift-Invariant Subspaces of Mixed Lebesgue Spaces

In this paper, the problem of reconstruction of signals in mixed Lebesgue spaces from their random average samples has been studied. Probabilistic sampling inequalities for certain subsets of shift-invariant spaces have been derived. It is shown that the probabilities increase to one when the sample size increases. Further, explicit reconstruction formulae for signals in these subsets have been obtained for which the numerical simulations have also been performed.

Parabolic equations with unbounded lower-order coefficients in Sobolev spaces with mixed norms

We prove the \(L_{p,q}\)-solvability of parabolic equations in divergence form with full lower-order terms. The coefficients and non-homogeneous terms belong to mixed Lebesgue spaces with the lowest integrability conditions. In particular, the coefficients for the lower-order terms are not necessarily bounded. We study both the Dirichlet and conormal derivative boundary value problems on irregular domains. We also prove embedding results for parabolic Sobolev spaces, the proof of which is of independent interest.

Stochastic integration with respect to fractional processes in Banach spaces

In the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is treated. The family of fractional processes that is considered includes, for example, fractional Brownian motions of any Hurst parameter or, more generally, fractionally filtered generalized Hermite processes. The class of Banach spaces that is considered includes a large variety of the most commonly used function spaces such as the Lebesgue spaces, Sobolev spaces, or, more generally, the Besov and Lizorkin-Triebel spaces. In the article, a characterization of the domains of the Wiener integrals on both bounded and unbounded intervals is given for both scalar and cylindrical fractional processes. In general, the integrand takes values in the space of γ-radonifying operators from a certain homogeneous Sobolev-Slobodeckii space into the considered Banach space. Moreover, an equivalent characterization in terms of a pointwise kernel of the integrand is also given if the considered Banach space is isomorphic with a subspace of a cartesian product of mixed Lebesgue spaces. The results are subsequently applied to stochastic convolution for which both necessary and sufficient conditions for measurability and sufficient conditions for continuity are found. As an application, space-time continuity of the solution to a parabolic equation of order 2m with distributed noise of low time regularity is shown as well as measurability of the solution to the heat equation with Neumann boundary noise of higher regularity.

Average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue space L^p,q(ℝ^d+1)

Average sampling and reconstruction in shift-invariant subspaces of mixed Lebesgue space L^p,q(ℝ^d+1)

The random convolution sampling stability in multiply generated shift invariant subspace of weighted mixed Lebesgue space

<abstract><p>In this paper, we mainly investigate the random convolution sampling stability for signals in multiply generated shift invariant subspace of weighted mixed Lebesgue space. Under some restricted conditions for the generators and the convolution function, we conclude that the defined multiply generated shift invariant subspace could be approximated by a finite dimensional subspace. Furthermore, with overwhelming probability, the random convolution sampling stability holds for signals in some subset of the defined multiply generated shift invariant subspace when the sampling size is large enough.</p></abstract>

Navier–Stokes regularity criteria in sum spaces

In this paper, we will consider regularity criteria for the Navier--Stokes equation in mixed Lebesgue sum spaces. In particular, we will prove regularity criteria that only require control of the velocity, vorticity, or the positive part of the second eigenvalue of the strain matrix, in the sum space of two scale critical spaces. This represents a significant step forward, because each sum space regularity criterion covers a whole family of scale critical regularity criteria in a single estimate. In order to show this, we will also prove a new inclusion and inequality for sum spaces in families of mixed Lebesgue spaces with a scale invariance that is also of independent interest.

On Function Spaces with Mixed Norms — A Survey

The targets of this article are threefold. The first one is to give a survey on the recent developments of function spaces with mixed norms, including mixed Lebesgue spaces, iterated weak Lebesgue spaces, weak mixed-norm Lebesgue spaces and mixed Morrey spaces as well as anisotropic mixed-norm Hardy spaces. The second one is to provide a detailed proof for a useful inequality about mixed Lebesgue norms and the Hardy--Littlewood maximal operator and also to improve some known results on the maximal function characterizations of anisotropic mixed-norm Hardy spaces and the boundedness of Calderon--Zygmund operators from these anisotropic mixed-norm Hardy spaces to themselves or to mixed Lebesgue spaces. The last one is to correct some errors and seal some gaps existing in the known articles.

Average sampling and reconstruction for signals in shift‐invariant subspaces of weighted mixed Lebesgue spaces

In this paper, we mainly study the average sampling problem for signals in a shift‐invariant subspace of weighted mixed Lebesgue space. First, the sampling stability for two kinds of average sampling functionals is established. Then, two iterative reconstruction algorithms with exponential convergence are presented for recovering the shift‐invariant signals from the corresponding average samples.

Frame-based Average Sampling in Multiply Generated Shift-invariant Subspaces of Mixed Lebesgue Spaces

In this paper, we mainly discuss the nonuniform average sampling and reconstruction in multiply generated shift-invariant subspaces \[ V_{p,q}(\Phi_r) = \bigg\{ \sum_{k_{1} \in \mathbf{Z}} \sum_{k_{2} \in \mathbf{Z}^{d}} c^T(k_{1},k_{2}) \Phi_r(\,\cdot-k_{1},\,\cdot-k_{2}): (c(k_{1},k_{2}))_{(k_{1},k_{2}) \in \mathbf{Z} \times \mathbf{Z}^{d}} \in \big( \ell^{p,q}(\mathbf{Z} \times \mathbf{Z}^d) \big)^r \bigg\} \] of mixed Lebesgue spaces $L^{p,q}(\mathbf{R} \times \mathbf{R}^{d})$, $1 \leq p,q \leq \infty$, where $\Phi_r = (\varphi_1, \varphi_2, \ldots, \varphi_r)^T$ with $\varphi_i \in L^{p,q}(\mathbf{R} \times \mathbf{R}^d)$ and $c = (c_1,c_2,\ldots,c_r)^T$ with $c_i \in \ell^{p,q}(\mathbf{Z} \times \mathbf{Z}^d)$, $i = 1,2,\ldots,r$, under the assumption that the family $\{ \varphi_{i}(x-k_{1},y-k_{2}): (k_{1},k_{2}) \in \mathbf{Z} \times \mathbf{Z}^{d}, 1 \leq i \leq r \}$ constitutes a $(p,q)$-frame of $V_{p,q}(\Phi_r)$. First, iterative approximation projection algorithms for two kinds of average sampling functionals are established. Then, we estimate the convergence rates of the corresponding algorithms.

Random sampling in multiply generated shift-invariant subspaces of mixed Lebesgue spaces [formula omitted

We mainly study the random sampling and reconstruction in multiply generated shift-invariant subspaces Vp,q(Φ) of mixed Lebesgue spaces Lp,q(R×Rd). Under suitable conditions for the generators Φ, we can prove that if the sampling sizes are large enough for both variables, the sampling stability holds with high probability for all functions in Vp,q(Φ) whose energy is concentrated on a compact subset. Finally, a reconstruction algorithm based on random samples is given for functions in a finite dimensional subspace.

Atomic Decomposition for Mixed Morrey Spaces

In this paper, we consider some norm estimates for mixed Morrey spaces considered by the first author. Mixed Lebesgue spaces are realized as a special case of mixed Morrey spaces. What is new in this paper is a new norm estimate for mixed Morrey spaces that is applicable to mixed Lebesgue spaces as well. An example shows that the condition on parameters is optimal. As an application, the Olsen inequality adapted to mixed Morrey spaces can be obtained.

Average sampling theorem for the homogeneous random fields in a reproducing kernel subspace of mixed Lebesgue space

In this paper, we mainly investigate the average sampling problem for the homogeneous random fields in a reproducing kernel subspace of mixed Lebesgue space. Based on the counterpart sampling result for the deterministic signals in the same space, a mean square convergence result for recovering the homogeneous random fields by the iterative reconstruction algorithm is obtained.