Search Results (90)

Let $0<q<1<p<\infty$ and let $\mathbb{T}$ be an arbitrary time scale. We establish a necessary-and-sufficient two-weight criterion in the quasi-Banach range for non-negative measurable functions from $L^p({[a,\infty )}_{\mathbb{T}},{\nu }^p \Delta t)$ to $L^q({[a,\infty )}_{\mathbb{T}},{\omega }^q \Delta t)$. The criterion is the finiteness of a mixed head–tail quantity involving $U\left(t\right)={\int }_t^{\infty }\omega {\left(\tau \right)}^q \Delta \tau$ and $V\left(\sigma \left(t\right)\right)={\int }_a^{\sigma \left(t\right)}\nu {\left(\tau \right)}^{-p^{\prime }} \Delta \tau$, and explicit two-sided bounds are obtained for the optimal constant. The key technical ingredient is a gap-compatible weighted level-function construction based on interval averages over right-scattered gaps. The result recovers the continuous characterization of Sinnamon and the discrete characterization of Braverman and Stepanov, and it also yields dynamic averaging, Bennett–Copson-type, Hardy–Flett-type, and shifted quantum specializations. Full article

We investigate the commutativity of Toeplitz operators with trigonometric polynomial symbols on the Beurling subspaces $z^k{H}_s^2$ of the Hardy–Sobolev spaces. An explicit formula for the commutator of two Toeplitz operators is first derived in terms of weighted shift operators. As a consequence, every such commutator is shown to have finite rank. It is then proved that if the degrees of the symbols are at most k, then the corresponding Toeplitz operators commute on the Beurling subspace $z^k{H}_s^2$. Moreover, this result is shown to be sharp by proving that, in general, the Beurling subspace $z^k{H}_s^2$ cannot be replaced by $z^{k-1}{H}_s^2$. Finally, these results generalize the corresponding commutativity theorem for Toeplitz operators on the classical Hardy space. Full article

We establish explicit direct and inverse approximation estimates for biorthogonal multiresolution projections on weighted Besov spaces over Muckenhoupt A_p weights. A Jackson-type direct estimate bounds the weighted L^p projection error by 2^−Js times the weighted Besov norm, and a matched Bernstein-type inverse estimate bounds the weighted Besov seminorm of a resolution-space element by 2^Js times its weighted L^p norm. Every constant is displayed in factorized form: each factor is either given in closed form or is the operator norm of the Hardy–Littlewood maximal operator on the weighted Lebesgue space, through which the entire dependence on the Muckenhoupt characteristic is routed. For a piecewise regular class combining a Besov-smooth component with finitely many net-zero jumps, the projection error separates into a smooth part decaying at 2^−Js and a localized jump part carrying the weighted measure of a shrinking interval about each jump; when the weight is locally comparable to Lebesgue measure near the jumps, this yields the effective rate $\min (s,1/p)$. A deterministic numerical experiment confirms this rate within one fixed biorthogonal analysis, and the same projection is illustrated on an empirical higher-education finance series. Full article

We establish a metrization result for some $b_v\left(s\right)$-metric spaces, extending a result recently established for rectangular b-metric spaces. Furthermore, we introduce the notion of an extended polygonal b-metric space (or $b_v\left(\theta \right)$-metric space), which unifies and generalizes several classes of spaces, including metric spaces, rectangular metric spaces, b-metric spaces, rectangular b-metric spaces, polygonal metric spaces, and $b_v\left(s\right)$-metric spaces. Some fixed-point results in $b_v\left(\theta \right)$-metric spaces are established under the weak orbital completeness condition in the framework of the Banach contraction principle and for generalized expansive Hardy-Rogers-type mappings. An a priori error estimate for the iterative process is obtained in both $b_v\left(\theta \right)$-metric and $b_v\left(s\right)$-metric spaces. We also establish the Ulam-Hyers stability of fixed-point equations in both $b_v\left(\theta \right)$-metric and $b_v\left(s\right)$-metric spaces. Several examples are provided, and applications to certain types of integral equations and initial value problems are presented, illustrating the applicability and effectiveness of the obtained results. Full article

This manuscript establishes the boundedness results for a class of singular integral operators to a more general framework by imposing conditions on the Hardy–Littlewood maximal function. We establish the results where we construct a suitable range space $L^{t(·)}\left({\mathbb{R}}_{+}\right)$ for a given domain space $L^{s(·)}\left({\mathbb{R}}_{+}\right)$ such that the operator maps $L^{s(·)}\left({\mathbb{R}}_{+}\right)$ into $L^{t(·)}\left({\mathbb{R}}_{+}\right)$ under appropriate assumptions. Conversely, for a prescribed range space $L^{t(·)}\left({\mathbb{R}}_{+}\right)$, we determine a corresponding domain space $L^{s(·)}\left({\mathbb{R}}_{+}\right)$ ensuring that the operator maps $L^{s(·)}\left({\mathbb{R}}_{+}\right)$ into $L^{t(·)}\left({\mathbb{R}}_{+}\right)$. In both settings, we provide illustrative examples to explicitly construct the respective spaces. Since our approach fundamentally relies on the boundedness of the maximal operator, the main results are valid only when the essential infimum $s_{-}>1$. In the limiting case $s_{-}=1$, we derive weak-type boundedness results of the form $(1,t(·\left)\right)$. Additionally, we present analogous formulations of these results in the classical setting. Full article

Let $\mathcal{A}=\{p+Jq:p,q\in \mathbb{C},J^2=-1\}$ be the commutative bicomplex-type algebra in which J commutes with the scalar imaginary unit. A Cauchy–Riemann-type operator $\overline{D}$ is studied on domains in ${\mathbb{C}}^2$. In the active coordinates $\xi =z_1-i{z}_2$ and $\eta =z_1+i{z}_2$, the equation $\overline{D}f=0$ is diagonal in the idempotent basis: the $e_{+}$-component is holomorphic in $\xi$ with $\eta$ as the parameter, while the $e_{-}$-component is holomorphic in $\eta$ with $\xi$ as the parameter. The expression $e_{+}F\left(\xi \right)+e_{-}G\left(\eta \right)$ is the parameter-independent subcase. From this decomposition, one obtains a slice characterization, a criterion for separatedness, a comparison with ordinary holomorphic functions of two complex variables, active-variable Cauchy formulas and estimates, local series with parameter-dependent coefficients, reflection symmetry, and Hardy and Bergman kernel lifts on the separated Hilbert spaces. Full article

In this manuscript, we introduce a class of measurable functions $\mathcal{A}\left({\mathbb{R}}_{+}\right)$, which is utilized to construct a generalized Bessel–Riesz kernel and the corresponding generalized Bessel–Riesz operator. We establish sufficient conditions ensuring that the generalized kernel belongs to variable Lebesgue spaces and derive a pointwise estimate for the associated operator in terms of the variable Lebesgue norm and the Hardy–Littlewood maximal operator. The main results provide boundedness criteria for the generalized Bessel–Riesz operator under appropriate assumptions on the exponent functions, as well as in more general settings where these conditions are relaxed. Furthermore, we demonstrate that boundedness results available in the literature can be recovered as special cases of our framework. In addition, we present an example that lies beyond the scope of existing results, thereby illustrating the wider applicability of our approach. In particular, when the exponent functions are constant, our results reduce to the classical Lebesgue space setting. Overall, this work extends and unifies a range of known results and provides a flexible framework for further developments in operator theory on variable Lebesgue spaces. Full article

We introduce the abstract weighted Morrey spaces $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )$, where $\mu$ is a general measure; investigate the properties of their predual spaces; and prove the boundedness of the Hardy–Littlewood maximal operator. Furthermore, we obtain an extrapolation theorem on $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )$, and consequently establish the norm inequalities for bounded oscillation ($\mathfrak{B}O$) operators on $M_{p,u}^{\kappa }(X,\mathfrak{M},\mu )$. As an application, we verify that $\mathfrak{B}O$ operators include the maximal operators, Littlewood–Paley square operators, and Carleson operators on classical weighted Morrey spaces, as well as Calderón–Zygmund operators on weighted Morrey spaces of homogeneous type. Full article

Within the present scholarly exposition, a meticulous and rigorous analysis is undertaken to investigate the boundedness characteristics of rough Hardy operators in the refined structural setting of variable exponent Morrey–Herz spaces. Moreover, parallel and quantitatively precise estimates are systematically established for the associated commutator operators, under the fundamental premise that the underlying symbol functions reside in the variable exponent bounded mean oscillation space $\left(BMO\right)$. Full article

This paper studies the algebraic properties of little Hankel operators on Hardy–Sobolev spaces $H_s^2$, focusing on a notion of commutativity defined via adjoint products. For symbols $\phi$ and $\psi$, whose co-analytic parts are trigonometric polynomials, we consider the condition ${\left(H_{\phi }^{\left(s\right)}\right)}^{*}{H}_{\psi }^{\left(s\right)}={\left(H_{\psi }^{\left(s\right)}\right)}^{*}{H}_{\phi }^{\left(s\right)}\mathrm{on}{H}_s^2.$ It is shown that this adjoint-product commutativity holds if and only if the co-analytic parts of the symbols are real scalar multiples of one another. As a consequence, the commutant of a nonzero Hankel operator on $H_s^2$, within the class of Hankel operators whose co-analytic symbols are trigonometric polynomials, is one-dimensional over $\mathbb{R}$. The proof relies on a direct coefficient analysis exploiting the finite Hankel structure induced by polynomial symbols. The result applies uniformly to all Sobolev exponents $s\ge 0$, including the classical Hardy space $s=0$ and the Dirichlet case $s=1/2$. Full article

We study fractional and complex powers of a fixed directional derivative in ${\mathbb{R}}^d$, defined via a Marchaud-type singular integral representation. Under explicit convergence assumptions, this yields a pointwise nonlocal realization along rays. We then formulate a Ramanujan–Hardy approach to fractional directional differentiation based on analytic interpolation of the directional jet at a point. This construction is local in jet space and is governed by Hardy’s formulation of Ramanujan’s Master Theorem. We emphasize that the resulting Ramanujan–Hardy derivative is defined through a Hardy-admissible interpolant of the directional jet. As an application, we investigate fractional directional derivatives of the Newtonian kernel in dimension $d\ge 3$. After a justified regularization and reduction to a Marchaud-type integral, we obtain a one-dimensional integral representation and a zonal harmonic description of the resulting function. This leads to a fractional Maxwell–Gegenbauer identity for $0<\Re \left(s\right)<1$, expressing the fractional directional derivative of ${\parallel x\parallel }^{2-d}$ in terms of Gegenbauer functions of complex degree. In this way, the classical Maxwell multipole formula appears as the integer-order case of a continuous analytic family. Moreover, the fractional operator preserves the main structural properties of the Newtonian kernel, including homogeneity, rotational invariance, and harmonicity away from the origin. The paper thus connects Mellin analysis, Ramanujan’s Master Theorem, fractional calculus, and harmonic analysis on the sphere, while clarifying the distinction between Marchaud and jet-interpolation constructions of fractional directional operators. Full article

In this paper, the uniform approximation of continuous functions on $[0,1]$ by rational functions with prescribed poles and bounded multiplicities is studied. A classical theorem of Fichera characterizes density in $C\left(\right[0,1\left]\right)$ through the divergence of a conformally invariant series involving the pole distribution. A modern reformulation of this result is developed and it is given an operator-theoretic interpretation in which the approximation property is equivalent to cyclicity and to the absence of nontrivial invariant subspaces in an associated Hardy-space model. In this framework, the classical Blaschke condition emerges as the fundamental obstruction to density, linking rational approximation to the structure of model spaces and non-selfadjoint operator algebras. The density criterion is interpreted in terms of symmetry: divergence corresponds to a balanced distribution of poles compatible with the conformal geometry of the slit domain, while convergence induces symmetry breaking and the emergence of invariant structures. Numerical models illustrate the sharpness of the criterion and provide a concrete manifestation of the Blaschke obstruction and cyclicity mechanism. This new approach places Fichera’s theorem within a broader operator-theoretic and spectral framework, connecting classical approximation theory with Hardy spaces, invariant subspace theory, and modern rational approximation methods. Full article

In this paper, we investigate the properties of the boundedness of fractional integral operators $K_{\alpha }$ defined on general measure metric spaces. We study their action in Lebesgue spaces $L^p\left(Y\right)$, Morrey spaces $L_{\phi }^p\left(Y\right)$, and extend our analysis to fractional Sobolev spaces $W^{\alpha ,p}\left(Y\right)$. Using classical dyadic decomposition and the Hardy–Littlewood maximal operator, we establish sharp bounds for $K_{\alpha }$ in terms of kernel parameters and the geometric structure of the space. A significant contribution of this work is the proof that $K_{\alpha }$ is bounded from $W^{\alpha ,p}\left(Y\right)$ to $L^q\left(Y\right)$, where thus linking our operator-theoretic framework with the theory of nonlocal and fractional partial differential equations. These results provide valuable tools for studying regularity, a priori estimates, and solution mappings in nonlocal problems involving the fractional Laplacian and related operators on irregular or non- Euclidean domains. Full article

Motivated by the fact that the Fibonacci sequence is the simplest nontrivial second-order recurrence with a rational generating function, we develop a Fibonacci-weighted Hardy theory for bicomplex holomorphic functions. Starting from the coefficient norm ${\sum }_{n\ge 0}{\left|a_n\right|}^2/F_{n+1}$, we obtain a bicomplex Hilbert module whose reproducing kernel is governed by ${(1-t-t^2)}^{-1}$ and whose maximal disk of holomorphy is determined sharply by the nearest kernel singularity, giving the radius ${\rho }_F={\phi }^{-1/2}$ (the square-root inverse of the golden ratio $\phi$). The arithmetic recurrence makes several objects fully explicit: we derive closed formulas for the kernels through the idempotent decomposition of $\mathbb{BC}$, compute exact norms of the shift powers and a golden-ratio spectral radius, and package the local theory into a sheaf of Fibonacci-holomorphic germs that are compatible with the bicomplex idempotent splitting. We also treat $(p,q)$-Fibonacci weights, obtaining a one-parameter family of rational kernels ${(1-pt-q{t}^2)}^{-1}$ and corresponding operator bounds. In addition to providing a concrete bicomplex model within weighted Hardy theory, the resulting explicit kernels furnish benchmark examples for kernel-based interpolation and for the operator theory of unilateral weighted shifts. Full article

We introduce and study a new class of nonlinear operators on metric spaces, called rational $(a,p)-$quasicontractions. Within this framework, we establish Greguš-type fixed-point theorems for closed, convex subsets of Banach spaces. The results establish the existence and uniqueness of fixed points, as well as the convergence of the Picard iteration for every initial guess. We show that rational $(a,p)-$quasicontractions strictly extend several classical contractive classes, including Hardy-Rogers, Kannan, Chatterjea, and rational contractions, and we provide explicit examples exhibiting the properness of these inclusions. As an application, we consider a nonlocal boundary value problem for a Caputo fractional differential equation of order $\alpha \in (1,2)$ with distributed delay and mixed nonlocal boundary conditions. By rewriting the problem as a Hammerstein-Volterra integral equation on a cone, and imposing natural growth and rational Lipschitz conditions on the delayed nonlinearity, we show that the associated Hammerstein operator is a rational $(a,p)-$quasicontraction. This yields the existence, uniqueness, and global attractivity of a positive solution. Two model fractional nonlinearities with delayed feedback are discussed in detail, along with a numerical scheme that illustrates the predicted geometric convergence of the discrete Picard iteration in the Caputo fractional setting. Full article