Our aim is to establish Sobolev type inequalities for fractional maximal functions $M_{ℍ,\nu }f$ and Riesz potentials $I_{ℍ,\alpha }f$ in weighted Morrey spaces of variable exponent on the half space $ℍ$. We also obtain Sobolev type inequalities for a $C^1$ function on $ℍ$. As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents $\Phi (x,t)=t^{p\left(x\right)}+{\left(b\left(x\right)t\right)}^{q\left(x\right)}$, where $p(·)$ and $q(·)$ satisfy log-Hölder conditions, $p\left(x\right)<q\left(x\right)$ for $x\in ℍ$, and $b(·)$ is nonnegative and Hölder continuous of order $\theta \in (0,1]$.

2000 Mathematics Subject Classification: 42B20, 42B25, 42B35We consider the generalized shift operator, generated by the Laplace- Bessel differential operator [...] The Bn -maximal functions and the Bn - Riesz potentials, generated by the Laplace-Bessel differential operator ∆Bn are investigated. We study the Bn - Riesz potentials in the Bn - Morrey spaces and Bn - BMO spaces. An inequality of Sobolev - Morrey type is established for the Bn - Riesz potentials.* This paper has been partially supported...

We prove two-weight norm estimates for fractional integrals and fractional maximal functions associated with starlike sets in Euclidean space. This is seen to include general positive homogeneous fractional integrals and fractional integrals on product spaces. We consider both weak type and strong type results, and we show that the conditions imposed on the weight functions are fairly sharp.

Let ${(X,\varrho ,\mu )}_{d,\theta }$ be a space of homogeneous type, i.e. X is a set, ϱ is a quasi-metric on X with the property that there are constants θ ∈ (0,1] and C₀ > 0 such that for all x,x’,y ∈ X, $|\varrho (x,y)-\varrho (x^{\text{'}},y)|\le C₀\varrho {(x,x^{\text{'}})}^{\theta }{[\varrho (x,y)+\varrho (x^{\text{'}},y)]}^{1-\theta }$, and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, $\mu \left(y\in X:\varrho (x,y)<r\right)\sim r^d$. Let ε ∈ (0,θ], |s| < ε and maxd/(d+ε),d/(d+s+ε) < q ≤ ∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces $F_{\infty q}^s\left(X\right)$ and establishes their frame characterizations by first establishing a Plancherel-Pólya-type inequality...

The purpose of this paper is to describe a unified approach to proving vector-valued inequalities without relying on the full strength of weighted theory. Our applications include the Fefferman-Stein and Córdoba-Fefferman inequalities, as well as the vector-valued Carleson operator. Using this approach we also produce a proof of the boundedness of the classical bi-parameter multiplier operators, which does not rely on product theory. Our arguments are inspired by the vector-valued restricted type...

New norms for some distributions on spaces of homogeneous type which include some fractals are introduced. Using inhomogeneous discrete Calderón reproducing formulae and the Plancherel-Pólya inequalities on spaces of homogeneous type, the authors prove that these norms give a new characterization for the Besov and Triebel-Lizorkin spaces with p, q > 1 and can be used to introduce new inhomogeneous Besov and Triebel-Lizorkin spaces with p, q ≤ 1 on spaces of homogeneous type. Moreover, atomic...

We study $L^p$ norm convergence of Bochner-Riesz means $S_R^{\delta }f$ associated with certain non-negative differential operators. When the kernel $S_R^m(x,y)$ satisfies a weak estimate for large values of m we prove $L^p$ norm convergence of $S_R^{\delta }f$ for δ > n|1/p-1/2|, 1 < p < ∞, where n is the dimension of the underlying manifold.

Properties of a maximal function for vector-valued martingales were studied by the author in an earlier paper. Restricting here to the dyadic setting, we prove the equivalence between (weighted) $L^p$ inequalities and weak type estimates, and discuss an extension to the case of locally finite Borel measures on ℝⁿ. In addition, to compensate for the lack of an $L^{\infty }$ inequality, we derive a suitable BMO estimate. Different dyadic systems in different dimensions are also considered.

Some martingale analogues of Sawyer's two-weight norm inequality for the Hardy-Littlewood maximal function Mf are shown for the Doob maximal function of martingales.

We characterize the pairs of weights on ℝ for which the operators $M_{h,k}^{+}f\left(x\right)=su{p}_{c>x}h(x,c){ʃ}_x^{c}f\left(s\right)k(x,s,c)ds$ are of weak type (p,q), or of restricted weak type (p,q), 1 ≤ p < q < ∞, between the Lebesgue spaces with the coresponding weights. The functions h and k are positive, h is defined on $(x,c):x<c$, while k is defined on $(x,s,c):x<s<c$. If $h(x,c)={(c-x)}^{-\beta }$, $k(x,s,c)={(c-s)}^{\alpha -1}$, 0 ≤ β ≤ α ≤ 1, we obtain the operator $M_{\alpha ,\beta }^{+}f=su{p}_{c>x}1/{(c-x)}^{\beta }{ʃ}_x^{c}f\left(s\right)/{(c-s)}^{1-\alpha }ds$. For this operator, under the assumption 1/p - 1/q = α - β, we extend the weak type characterization to the case p = q and prove that in the case of equal weights and...