Space Lq Research Articles

Fractional Hermite–Hadamard, Newton–Milne, and Convexity Involving Arithmetic–Geometric Mean-Type Inequalities in Hilbert and Mixed-Norm Morrey Spaces ℓq(·)(Mp(·),v(·)) with Variable Exponents

Function spaces play a crucial role in the study and application of mathematical inequalities. They provide a structured framework within which inequalities can be formulated, analyzed, and applied. They allow for the extension of inequalities from finite-dimensional spaces to infinite-dimensional contexts, which is crucial in mathematical analysis. In this note, we develop various new bounds and refinements of different well-known inequalities involving Hilbert spaces in a tensor framework as well as mixed Moore norm spaces with variable exponents. The article begins with Newton–Milne-type inequalities for differentiable convex mappings. Our next step is to take advantage of convexity involving arithmetic–geometric means and build various new bounds by utilizing self-adjoint operators of Hilbert spaces in tensorial frameworks for different types of generalized convex mappings. To obtain all these results, we use Riemann–Liouville fractional integrals and develop several new identities for these operator inequalities. Furthermore, we present some examples and consequences for transcendental functions. Moreover, we developed the Hermite–Hadamard inequality in a new and significant way by using mixed-norm Moore spaces with variable exponent functions that have not been developed previously with any other type of function space apart from classical Lebesgue space. Mathematical inequalities supporting tensor Hilbert spaces are rarely examined in the literature, so we believe that this work opens up a whole new avenue in mathematical inequality theory.

Kolmogorov widths of an intersection of a family of balls in a mixed norm

In this paper, order estimates for the Kolmogorov n-widths of an intersection of a family of balls in a mixed norm in the space lq,σm,k with 2⩽q,σ<∞, n⩽mk/2 are obtained.

Strongly Lipschitz (ℓp ,ℓq)-factorable mappings

In this paper we study the space of strongly Lipschitz (ℓp ,ℓq) -factorable operators between metric spaces and a Banach spaces. In particular, a factorization of this class through ℓp and ℓq spaces is given. We show that this type of operators fits in the theory of composition α-Banach Lipschitz operator ideal. As a special case, we get a Lipschitz version of weakly p-nuclear operators.

A nonlinear elliptic equation with a degenerate diffusion and a source term in [formula omitted

We study a simplified equation governing turbulent kinetic energy k in a bounded domain, arising from turbulence modeling where the eddy diffusion is given by ϱ(x)+ɛ, with ϱ representing the Prandtl mixing length of the order of the distance to the boundary, and a right-hand side in L1. We obtain estimates of ϱ∇k in Lq spaces and we establish the convergence toward the formal limit equation in the sense of the distributions as ɛ goes to 0.

Limit theorems for multifractal products of random fields

This paper investigates asymptotic properties of multifractal products of random fields and multidimensional multifractal measures. The obtained limit theorems provide sufficient conditions for the convergence of cumulative fields in the spaces Lq. New results on the rate of convergence of cumulative fields are presented. Simple unified conditions for the limit theorems and the calculation of the Rényi function are given. They are less restrictive than those in the known one-dimensional results. The developed methodology is also applied to multidimensional multifractal measures. Finally, an application to a new class of examples of geometric φ-sub-Gaussian random fields is presented. In this case, the general assumptions have a simple form and can be expressed in terms of covariance functions only.

Existence of solutions to fluid equations in Hölder and uniformly local Sobolev spaces

We establish short-time existence of solutions to the surface quasi-geostrophic (SQG) equation in the Hölder spaces Cr(R2) for r>1; to avoid an integrability assumption (such as membership of the data in an Lq space) we introduce a generalization of the SQG constitutive law. As an application of the Hölder theory, we use these solutions when forming an approximation sequence in the proof of existence of solutions of SQG in another class of non-decaying function spaces, the uniformly local Sobolev spaces Huls(R2) for s≥3. Using methods similar to those for the surface quasi-geostrophic equation, we also obtain short-time existence for the three-dimensional Euler equations in uniformly local Sobolev spaces.

SDEs with critical time dependent drifts: Weak solutions

For d≥3, we prove that time-inhomogeneous stochastic differential equations driven by additive noises with drifts in critical Lebesgue space Lq([0,T];Lp(Rd)), where (p,q)∈(d,∞]×[2,∞) and d∕p+2∕q=1, or (p,q)=(d,∞) and divb∈L∞([0,T];Ld∕2+ε(Rd)), are well-posed. The weak uniqueness is obtained by solving corresponding Kolmogorov backward equations in some second-order Sobolev spaces, which is analytically interesting in itself.

Identifying of unknown source term for the Rayleigh-Stokes problem

In this paper, we would like to briefly introduce some applications of fractional derivatives in the fields of heat and fluid-flows. However, our main focus is on study an inverse source problem for the Rayleigh-Stokes problem. The problem is severely ill-posed. We verify the ill-posedness of Problem 1, applying the modified Lavrentiev to construct a regularization from the exact data. After that, we established the convergent rate between the exact solution and its approximation. Furthermore, we have the estimate in Lq space.

Analytical and numerical dissipativity for the space-fractional Allen–Cahn equation

This paper is concerned with the analytical and numerical dissipativity of the space-fractional Allen–Cahn equation, a generalization of the classic Allen–Cahn equation by replacing the local Laplacian with a nonlocal fractional Laplacian. It is first proved that the continuous dynamical system is dissipative as its local counterpart in Hα and Lq, q=2k+2 for k≥0, spaces. Then it is shown that the backward Euler method preserves the dissipativity of the underlying system, that is, the discrete-in-time dynamical system with time-step parameter τ is still dissipative in Hα and Lq spaces. The existence of the global attractor for both continuous and discrete dynamical systems are then obtained. A numerical example is given to confirm the theoretical results.

Bohr and Rogosinski inequalities for operator valued holomorphic functions

For any complex Banach space X and each p∈[1,∞), we introduce the p-Bohr radius of order N(∈N) is R˜p,N(X) defined byR˜p,N(X)=sup⁡{r≥0:∑k=0N‖xk‖prpk≤‖f‖H∞(D,X)p}, where f(z)=∑k=0∞xkzk∈H∞(D,X). Here D={z∈C:|z|<1} denotes the unit disk. We also introduce the following geometric notion of p-uniformly C-convexity of order N for a complex Banach space X for some N∈N. For p∈[2,∞), a complex Banach space X is called p-uniformly C-convex of order N if there exists a constant λ>0 such that(0.1)(‖x0‖p+λ‖x1‖p+λ2‖x2‖p+⋯+λN‖xN‖p)1/p≤maxθ∈[0,2π)⁡‖x0+∑k=1Neiθxk‖ for all x0, x1, …, xN ∈X. We denote Ap,N(X), the supremum of all such constants λ satisfying (0.1). We obtain the lower and upper bounds of R˜p,N(X) in terms of Ap,N(X). In this paper, for p∈[2,∞) and each N∈N, we prove that a complex Banach space X is p-uniformly C-convex of order N if, and only if, the p-Bohr radius of order NR˜p,N(X)>0. We also study the p-Bohr radius of order N for the Lebesgue spaces Lq(μ) for 1≤p<q<∞ or 1≤q≤p<2. Finally, we prove an operator valued analogue of a refined version of Bohr and Rogosinski inequality for bounded holomorphic functions from the unit disk D into B(H), where B(H) denotes the space of all bounded linear operator on a complex Hilbert space H.

Local Hölder and maximal regularity of solutions of elliptic equations with superquadratic gradient terms

We study the local Hölder regularity of strong solutions u of second-order uniformly elliptic equations having a gradient term with superquadratic growth γ>2, and right-hand side in a Lebesgue space Lq. When q>Nγ−1γ (N is the dimension of the Euclidean space), we obtain the optimal Hölder continuity exponent αq>γ−2γ−1. This allows us to prove some new results of maximal regularity type, which consist in estimating the Hessian matrix of u in Lq. Our methods are based on blow-up techniques and a Liouville theorem.

On some Lambda–Pascal sequence spaces and compact operators

We introduce Lambda–Pascal sequence spaces ℓq(G), c0(G), c(G) and ℓ∞(G) generated by the matrix G which is obtained by the product of Pascal matrix and Λ-matrix. It is proved that the Lambda–Pascal sequence spaces ℓq(G), c0(G), c(G) and ℓ∞(G) are BK-spaces and linearly isomorphic to ℓq, c0, c and ℓ∞, respectively. We construct Schauder bases and obtain α-, β- and γ-duals of the new spaces. We state and prove characterization theorems related to matrix transformation from the space ℓq(G) to the spaces ℓ∞, c and c0. Finally, we determine necessary and sufficient conditions for a matrix operator to be compact from the space c0(G) to any one of the spaces ℓ∞, c, c0 or ℓ1.

Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted Lq spaces on metric spaces

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into Lq spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincaré type inequalities. We show how to construct such suitable coverings and Poincaré inequalities. For locally doubling measures, we prove a self-improvement property for two-weighted Poincaré inequalities, which applies also to lower-dimensional measures.We simultaneously treat various Sobolev spaces, such as the Newtonian, fractional Hajłasz and Poincaré type spaces, for rather general measures and sets, including fractals and domains with fractal boundaries. By considering lower-dimensional measures on the boundaries of such domains, we obtain trace embeddings for the above spaces. In the case of Newtonian spaces we exactly characterize when embeddings into Lq spaces with respect to another measure are compact. Our tools are illustrated by concrete examples. For measures satisfying suitable dimension conditions, we recover several classical embedding theorems on domains and fractal sets in Rn.

Bilinear θ-type generalized fractional integral operator and its commutator on some non-homogeneous spaces

Let (X,d,μ) be a non-homogeneous metric measure space which satisfies the upper doubling and geometrically doubling conditions in the sense of Hytönen. Under assumption that the dominating function λ satisfies weak reverse doubling condition, the authors prove that bilinear θ-type generalized fractional integral BT˜α,θ is bounded from the product of spaces Lp1(μ)×Lp2(μ) into space Lq(μ), and bounded from the product of spaces Mq1p1(μ)×Mq2p2(μ) into space Mqp(μ). Moreover, via establishing the sharp maximal estimate for the commutator [b1,b2,BT˜α,θ] which is generated by b1,b2∈RBMO˜(μ) and BT˜α,θ, the boundedness of [b1,b2,BT˜α,θ] on space Lp(μ) and on space Mqp(μ) is also obtained.

Recovering the initial distribution for a logarithmic nonlinear biparabolic equation

In this paper, we consider the problem of finding the initial distribution for a biparabolic equation with logarithmic nonlinearity source. The problem is severely ill‐posed in the sense of Hadamard. Under the a priori assumption on the exact solution belonging to a Gevrey space, we consider a generalized nonlinear problem by using the Fourier truncation method to obtain rigorous convergence estimates in the norms on Lq space with some q ≥ 2 and the Sobolev space Ws, ℓ for some constants s &gt; 0, ℓ ≥ 1.

Weighted Fractional Hardy Operators and their Commutators on Generalized Morrey Spaces over Quasi-Metric Measure Spaces

Abstract We study commutators of weighted fractional Hardy-type operators within the frameworks of local generalized Morrey spaces over quasi-metric measure spaces for a certain class of “radial” weights. Quasi-metric measure spaces may include, in particular, sets of fractional dimentsions. We prove theorems on the boundedness of commutators with CMO coefficients of these operators. Given a domain Morrey space 𝓛 p,φ (X) for the fractional Hardy operators or their commutators, we pay a special attention to the study of the range of the exponent q of the target space 𝓛 q,ψ (X). In particular, in the case of classical Morrey spaces, we provide the upper bound of this range which is greater than the known Adams exponent.

Lower bounds of cowidths and widths of multiplier operators

The main objective of this article is to present new results on optimal reconstruction of function classes on probability spaces (Ω,A,ν) in the standard Lq spaces. We consider the problem of optimal reconstruction in the sense of the respective cowidths of standard function classes ΛUp generated by multiplier or pseudo differential operators Λ:Lp→Lq, 1≤p,q≤∞. Our approach is based on the estimates of volumes of John-Löwner ellipsoids and expectations of norms induced by orthonormal systems on (Ω,A,ν). It is shown that the results obtained are order sharp in many cases. In particular, we obtain sharp orders of entropy of Sobolev classes W∞γ, γ>0 in L1 and n-widths of ΛUp in Lq, 1<q≤p<∞ in the case of two-point homogeneous spaces and torus.

An extension of Calderón-Zygmund type singular integral with non-smooth kernel

In the present paper, we consider a kind of singular integralTf(x)=p.v.∫RnΩ(y)|y|n−βf(x−y)dy which can be viewed as an extension of the classical Calderón-Zygmund type singular integral. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish an estimate of the singular integral in the Lq space for 1<q<∞ and a weak (1,1) type of the singular integral when 0<β<(q−1)nq without any smoothness assumed on Ω. Moreover, the bounds do not depend on β and the strong (q,q) type estimate and weak (1,1) type estimate of the Calderón-Zygmund type singular integral can be recovered when β→0 from our obtained estimates.

An anisotropic regularity condition for the 3D incompressible Navier–Stokes equations for the entire exponent range

We show that a suitable weak solution to the incompressible Navier–Stokes equations on R3×(−1,1) is regular on R3×(−1,0] if ∂3u belongs to M2p/(2p−3),α((−1,0);Lp(R3)) for any α>1 and p∈(3/2,∞), which is a logarithmic-type variation of a Morrey space in time. For each α>1 this space is, up to a logarithm, critical with respect to the scaling of the equations, and contains all spaces Lq((−1,0);Lp(R3)) that are subcritical, that is for which 2/q+3/p<2.

Intersections of ℓp spaces in the Borel hierarchy

We show that if Y is one of the spaces ℓq, c0, ℓ∞ or ⋂p>bℓp where 0<q,b<∞, and if the F-space ⋂p>aℓp is contained in Y properly (0⩽a<∞), then ⋂p>aℓp first shows up in the Borel hierarchy of Y at the multiplicative class of the third level. In particular ⋂p>aℓp is neither an Fσ nor a Gδ subset of Y. This answers a question by Nestoridis. This result provides a natural example of a set in the third level of the Borel hierarchy and with its help we also give some examples in the fourth level.