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In this paper, we prove the boundedness of the Bn maximal operator and Bn singular integral operators associated with the Laplace-Bessel differential operator Δ Bn on variable exponent Lebesgue spaces.

We obtain the global weighted Morrey-type regularity of the solution of the regular oblique derivative problem for linear uniformly parabolic operators with VMO coefficients. We show that if the right-hand side of the parabolic equation belongs to certain generalized weighted Morrey space Mp,ϕ (Q, w), than the strong solution belongs to the generalized weighted Sobolev- Morrey space W ˙ 2 , 1 p , φ ( Q , ω ) $\dot W_{2,1}^{p,\varphi }\left( {Q,\omega } \right)$ .

In this paper, we study the Riesz potential (G-Riesz potential) generated by the Gegenbauer differential operator G λ = ( x 2 - 1 ) 1 2 - λ ⁢ d d ⁢ x ⁢ ( x 2 - 1 ) λ + 1 2 ⁢ d d ⁢ x , x ∈ ( 1 , ∞ ) , λ ∈ ( 0 , 1 2 ) . G_{\lambda}=(x^{2}-1)^{\frac{1}{2}-\lambda}\frac{d}{dx}(x^{2}-1)^{\lambda+% \frac{1}{2}}\frac{d}{dx},\quad x\in(1,\infty),\,\lambda\in\Bigl{(}0,\frac{1}{2% }\Bigr{)}. We prove that the G-Riesz potential I G α {I_{G}^{\alpha}} , 0 < α < 2 ⁢ λ + 1 {0 , is bounded from the G-Morrey space L p , λ , γ {L_{p,\lambda,\gamma}} to L q , λ , γ {L_{q,\lambda,\gamma}} if and only if 1 p - 1 q = α 2 ⁢ λ + 1 - γ , 1 < p < 2 ⁢ λ + 1 - γ α . \frac{1}{p}-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma},\quad 1 Also, we prove that the G-Riesz potential I G α {I_{G}^{\alpha}} is bounded from the G-Morrey space L 1 , λ , γ {L_{1,\lambda,\gamma}} to the weak G-Morrey space W ⁢ L q , λ , γ {WL_{q,\lambda,\gamma}} if and only if 1 - 1 q = α 2 ⁢ λ + 1 - γ . 1-\frac{1}{q}=\frac{\alpha}{2\lambda+1-\gamma}.

In this paper we study the fractional maximal operator Mα, 0 ≤ α < Q and the Riesz potential operator ℑα, 0 < α < Q on the Heisenberg group in the modified Morrey spaces L͂p,λ(ℍn), where Q = 2n + 2 is the homogeneous dimension on ℍn. We prove that the operators Mα and ℑα are bounded from the modified Morrey space L͂1,λ(ℍn) to the weak modified Morrey space WL͂q,λ(ℍn) if and only if, α/Q ≤ 1 - 1/q ≤ α/(Q - λ) and from L͂p,λ(ℍn) to L͂q,λ(ℍn) if and only if, α/Q ≤ 1/p - 1/q ≤ α/(Q - λ).In the limiting case we prove that the operator Mα is bounded from L͂p,λ(ℍn) to L∞(ℍn) and the modified fractional integral operator Ĩα is bounded from L͂p,λ(ℍn) to BMO(ℍn).As applications of the properties of the fundamental solution of sub-Laplacian Ը on ℍn, we prove two Sobolev-Stein embedding theorems on modified Morrey and Besov-modified Morrey spaces in the Heisenberg group setting. As an another application, we prove the boundedness of ℑα from the Besov-modified Morrey spaces BL͂spθ,λ(ℍn) to BL͂spθ,λ(ℍn).