Search Results (17)

This paper investigates a new class of fractional integral operators, namely, the exponentially damped Riesz-type operators within the framework of variable exponent Lebesgue spaces $L^{\mathtt{p}(·)}$. To the best of our knowledge, the boundedness of such operators has not been addressed in any existing functional setting. We establish their boundedness under appropriate log-Hölder continuity and growth conditions on the exponent function $\mathtt{p}(·)$. To highlight the novelty and practical relevance of the proposed operator, we conduct a comparative analysis demonstrating its effectiveness in addressing convergence, regularity, and stability of solutions to partial differential equations. We also provide non-trivial examples that illustrate not only these properties but also show that, under this operator, a broader class of functions becomes locally integrable. The exponential decay factor notably broadens the domain of boundedness compared to classical Riesz and Bessel–Riesz potentials, making the operator more versatile and robust. Additionally, we generalize earlier results on Sobolev-type inequalities previously studied in constant exponent spaces by extending them to the variable exponent setting through our fractional operator, which reduces to the classical Riesz potential when the decay parameter $\lambda =0$. Applications to elliptic PDEs are provided to illustrate the functional impact of our results. Furthermore, we develop several new structural properties tailored to variable exponent frameworks, reinforcing the strength and applicability of the proposed theory. Full article

This study demonstrates several novel dynamic inequalities of the Hardy and Littlewood types on time scales. As special cases, our studies include Hardy’s integral inequalities and Hardy and Littlewood’s discrete inequalities. The research findings are demonstrated using algebraic inequalities, Hölder’s inequality, and the chain rule on time scales. Full article

This paper investigates the Bessel–Riesz operator within the framework of variable Lebesgue spaces. We extend existing results by establishing boundedness under more general conditions. The analysis is based on the Hardy–Littlewood maximal function, Hölder’s inequality, and dyadic decomposition techniques. For a given domain space, we construct a suitable range space such that the operator remains bounded. Conversely, for a prescribed range space, we identify a corresponding domain space that guarantees boundedness. Illustrative examples are included to demonstrate the construction of such spaces. The main results hold when the essential infimum of the exponent function exceeds one, and we also establish weak-type estimates in the limiting case. Full article

In this manuscript, we establish the boundedness of the Bessel–Riesz operator $I_{\alpha ,\gamma }f$ in variable Lebesgue spaces $L^{p(·)}$. We prove that $I_{\alpha ,\gamma }f$ is bounded from $L^{p(·)}$ to $L^{p(·)}$ and from $L^{p(·)}$ to $L^{q(·)}$. We explore various scenarios for the boundedness of $I_{\alpha ,\gamma }f$ under general conditions, including constraints on the Hardy–Littlewood maximal operator $\mathcal{M}$. To prove these results, we employ the boundedness of $\mathcal{M}$, along with Hölder’s inequality and classical dyadic decomposition techniques. Our findings unify and generalize previous results in classical Lebesgue spaces. In some cases, the results are new even for constant exponents in Lebesgue spaces. Full article

Let $M_g^{+}f$ be the one-sided Hardy–Littlewood maximal function, ${\phi }_1$ be a nonnegative and nondecreasing function on $[0,\infty )$, $\gamma$ be a positive and nondecreasing function defined on $[0,\infty )$; let ${\phi }_2$ be a quasi-convex function and $u,v,w$ be three weight functions. In this paper, we present necessary and sufficient conditions on weight functions $(u,v,w)$ such that the inequality ${\phi }_1\left(\lambda \right){\int }_{\{M_g^{+}f>\lambda \}}u\left(x\right)g\left(x\right)dx\le C{\int }_{-\infty }^{+\infty }{\phi }_2\left(C{\displaystyle \frac{\left|f\right(x\left)\right|v\left(x\right)}{\gamma \left(\lambda \right)}}\right)w\left(x\right)g\left(x\right)dx$ holds. Then, we unify the weak and extra-weak-type one-sided Hardy–Littlewood maximal inequalities in the above inequality. Full article

In this paper, by introducing multiple parameters, we establish a discrete version of the Hardy–Littlewood–Polya inequality in the whole plane. For the obtained inequality, we give the equivalent statements of the best possible constant factor linked to the parameters and deal with the equivalent inequalities. Our main result provided a new generalization of Hardy–Littlewood–Polya inequality, and as a consequence, we show that some new inequalities of the Hardy–Littlewood–Polya type can be derived from the current results by taking the special values of parameters. Full article

In this study, some new necessary and sufficient conditions for a two-weight, weak-type maximal inequality of the form ${\phi }_1\left(\lambda \right){\int }_{\{x\in X:Mf(x)>\lambda \}}\varrho \left(x\right)d\mu \left(x\right)\le c{\int }_X{\phi }_2\left(c\left|f\right(x\left)\right|\right)\sigma \left(x\right)d\mu \left(x\right)$ are obtained in Orlicz classes, where $Mf$ is a Hardy–Littlewood maximal function defined on homogeneous spaces and $\varrho$ is a weight function. Full article

This paper explores the properties of multipliers associated with discrete analogues of fractional integrals, revealing intriguing connections with Dirichlet characters, Euler’s identity, and Dedekind zeta functions of quadratic imaginary fields. Employing Fourier transform techniques, the Hardy–Littlewood circle method, and a discrete analogue of the Stein–Weiss inequality on product space through implication methods, we establish ${\ell }^p\to {\ell }^q$ bounds for these operators. Our results contribute to a deeper understanding of the intricate relationship between number theory and harmonic analysis in discrete domains, offering insights into the convergence behavior of these operators. Full article

As Fourier transformations of $L^p$ functions are the mathematical basis of various applications, it is necessary to develop $L^p$ theory for 2D-LCT before any further rigorous mathematical investigation of such transformations. In this paper, we study this $L^p$ theory for $1\le p<\infty$. By defining an appropriate convolution, we obtain a result about the inverse of 2D-LCT on $L^1\left({\mathbb{R}}^2\right)$. Together with the Plancherel identity and Hausdorff–Young inequality, we establish $L^p\left({\mathbb{R}}^2\right)$ multiplier theory and Littlewood–Paley theorems associated with the 2D-LCT. As applications, we demonstrate the recovery of the $L^1\left({\mathbb{R}}^2\right)$ signal function by simulation. Moreover, we present a real-life application of such a theory of 2D-LCT by encrypting and decrypting real images. Full article

Let $(\mathcal{X},d)$ be a directionally $(\gamma ,m)$-limited space with every $\gamma \in (0,\infty )$. In this setting, we aim to study an analogue of the classical theory of $A_p\left(\mu \right)$ weights. As an application, we establish some weighted estimates for the Hardy–Littlewood maximal operator. Then, we introduce the relationship between directionally $(\gamma ,m)$-limited spaceand geometric doubling. Finally, we obtain the weighted norm inequalities of the Calderón–Zygmund operator and commutator in non-homogeneous space. Full article

In recent years, the theory of convex mappings has gained much more attention due to its massive utility in different fields of mathematics. It has been characterized by different approaches. In 1929, G. H. Hardy, J. E. Littlewood, and G. Polya established another characterization of convex mappings involving an ordering relationship defined over ${\mathbb{R}}^{\mathfrak{n}}$ known as majorization theory. Using this theory many inequalities have been obtained in the literature. In this paper, we study Hermite–Hadamard type inequalities using the Jensen–Mercer inequality in the frame of $\underset{\dot{}}{\mathrm{q}}$-calculus and majorized l-tuples. Firstly we derive $\underset{\dot{}}{\mathrm{q}}$-Hermite–Hadamard–Jensen–Mercer (H.H.J.M) type inequalities with the help of Mercer’s inequality and its weighted form. To obtain some new generalized (H.H.J.M)-type inequalities, we prove a generalized quantum identity for $\underset{\dot{}}{\mathrm{q}}$-differentiable mappings. Next, we obtain some estimation-type results; for this purpose, we consider $\underset{\dot{}}{\mathrm{q}}$-identity, fundamental inequalities and the convexity property of mappings. Later on, We offer some applications to special means that demonstrate the importance of our main results. With the help of numerical examples, we also check the validity of our main outcomes. Along with this, we present some graphical analyses of our main results so that readers may easily grasp the results of this paper. Full article

The paper deals with studying a connection of the Littlewood–Offord problem with estimating the concentration functions of some symmetric infinitely divisible distributions. It is shown that the concentration function of a weighted sum of independent identically distributed random variables is estimated in terms of the concentration function of a symmetric infinitely divisible distribution whose spectral measure is concentrated on the set of plus-minus weights. Full article

In this paper, Hardy–Leindler, Hardy–Yang and Hwang type inequalities are extended on time scales calculus. These extensions are depending upon use of symmetric multiple delta integrals. The target is achieved by utilizing some inequalities in literature along with mathematical induction principle and Fubini’s theorem on time scales. The obtained inequalities are discussed in discrete, continuous and quantum calculus in search of applications. Particular cases of proved results include Hardy, Copson, Hardy–Littlewood, Levinson and Bennett-type inequalities for symmetric sums. Full article

In this paper, the authors establish the bounds for the Hardy-Littlewood maximal operator defined on a finite directed graph $\overrightarrow{G}$ in the space ${\mathrm{BV}}_p(\overrightarrow{G})$ of bounded p-variation functions. More precisely, the authors obtain the ${\mathrm{BV}}_p$ norms of $M_{\overrightarrow{G}}$ for some directed graphs $\overrightarrow{G}$. Full article

This study is based on a part of the results obtained in the author’s publications. An enumerable family of the Le Roy type functions is considered herein. The asymptotic formula for these special functions in the cases of ‘large’ values of indices, that has been previously obtained, is provided. Further, series defined by means of the Le Roy type functions are considered. These series are studied in the complex plane. Their domains of convergence are given and their behaviour is investigated ‘near’ the boundaries of the domains of convergence. The discussed asymptotic formula is used in the proofs of the convergence theorems for the considered series. A theorem of the Cauchy–Hadamard type is provided. Results of Abel, Tauber and Littlewood type, which are analogues to the corresponding theorems for the classical power series, are also proved. At last, various interesting particular cases of the discussed special functions are considered. Full article