Morrey Norm Research Articles

Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operators

Generalized Hölder estimates via generalized Morrey norms for some ultraparabolic operators

Existence of ground‐state solutions for p‐Choquard equations with singular potential and doubly critical exponents

AbstractWe are concerned with the following Choquard equation: where , , , , is the p‐Laplacian, is the Riesz potential, and F is the primitive of f which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. Under different range of θ and almost necessary conditions on the nonlinearity f in the spirit of Berestycki–Lions‐type conditions, we divide this paper into three parts. By applying the refined Sobolev inequality with Morrey norm and the generalized version of the Lions‐type theorem, some existence results are established. It is worth noting that our method is not involving the concentration–compactness principle.

A global compactness result with applications to a nonlinear elliptic equation arising in astrophysics

In this paper, we study a nonlinear elliptic equation arising in astrophysics:(0.1)−Δu+V(x)u=|u|2⁎−2u|x′|+Q(x)|u|q−2u,x:=(x′,x″)∈Rm×Rn−m, where n≥3, 2≤m<n, 2⁎=2(n−1)n−2, 2<q<2nn−2 and V(x),Q(x)∈C(Rn). The partly singular term |u|2⁎−2u|x′| in (0.1) forces us to establish a refined Sobolev inequality involving partly weighted Morrey norm, which generalizes similar inequality obtained by G. Palatucci and A. Pisante (2014) [25]. Benefiting from the new inequality, we obtain a global compactness result and some existence result for (0.1), which extend the results obtained by Y. B. Deng et al. (2012) [12]. Our strategy turns out to be more concise because we avoid the use of Levy concentration functions.

Ground state solutions and decay estimation of Choquard equation with critical exponent and Dipole potential

&lt;p style='text-indent:20px;'&gt;In this paper, we study a class of Choquard equations with critical exponent and Dipole potential. We prove the existence of radial ground state solutions for Choquard equations by using the refined Sobolev inequality with the Morrey norm, and show that any nonnegative weak solutions of Choquard equations have additional decay properties in terms of ground state representation.&lt;/p&gt;

Fractional Sobolev embedding with radial potential

We present some embedding results of fractional Ds,2(RN) space with radial potential. To overcome the difficulty of non-compactness, we obtain the Lions-type theorems by means of the refined Sobolev inequality with the Morrey norm. Then we apply the obtained results to study the existence of ground state solution for fractional equations with radial potential.

P-Laplacian equation with finitely many critical nonlinearities

This article concerns the ground state solution of the p-Laplacian equation with finitely many critical nonlinearities. By using the refined Sobolev inequality with Morrey norm and variational methods, we establish the existence of nonnegative ground state solution.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/102/abstr.html

The Equivalence of Operator Norm between the Hardy-Littlewood Maximal Function and Truncated Maximal Function on the Heisenberg Group

In this article, we define a kind of truncated maximal function on the Heisenberg space by M γ c f x = sup 0 &lt; r &lt; γ 1 / m B x , r ∫ B x , r f y d y . The equivalence of operator norm between the Hardy-Littlewood maximal function and the truncated maximal function on the Heisenberg group is obtained. More specifically, when 1 &lt; p &lt; ∞ , the L p norm and central Morrey norm of truncated maximal function are equal to those of the Hardy-Littlewood maximal function. When p = 1 , we get the equivalence of weak norm L 1 ⟶ L 1 , ∞ and M ̇ 1 , λ ⟶ W ̇ M 1 , λ . Those results are generalization of previous work on Euclid spaces.

On doubly critical coupled systems involving fractional Laplacian with partial singular weight

In this paper, we consider the doubly critical coupled systems involving fractional Laplacian in with partial singular weight: where s ∈ (0, 1), 0 ≤ α, β &lt; 2s &lt; n, 0 &lt; m &lt; n, , η1, η2 &gt; 1, , γ1, γ2 &lt; γH, and is some explicit constant. By establishing new embedding results involving partially weighted Morrey norms in the product space , we provide sufficient conditions under which a weak nontrivial solution of (0.1) exists via variational methods. We also extend these results to p‐Laplacian systems especially.

Boundedness of composition operators on Morrey spaces and weak Morrey spaces

In this study, we investigate the boundedness of composition operators acting on Morrey spaces and weak Morrey spaces. The primary aim of this study is to investigate a necessary and sufficient condition on the boundedness of the composition operator induced by a diffeomorphism on Morrey spaces. In particular, detailed information is derived from the boundedness, i.e., the bi-Lipschitz continuity of the mapping that induces the composition operator follows from the continuity of the composition mapping. The idea of the proof is to determine the Morrey norm of the characteristic functions, and employ a specific function composed of a characteristic function. As this specific function belongs to Morrey spaces but not to Lebesgue spaces, the result reveals a new phenomenon not observed in Lebesgue spaces. Subsequently, we prove the boundedness of the composition operator induced by a mapping that satisfies a suitable volume estimate on general weak-type spaces generated by normed spaces. As a corollary, a necessary and sufficient condition for the boundedness of the composition operator on weak Morrey spaces is provided.

On limiting trace inequalities for vectorial differential operators

We establish that trace inequalities $$\|D^{k-1}u\|_{L^{\frac{n-s}{n-1}}(\mathbb{R}^{n},d\mu)} \leq c \|\mu\|_{L^{1,n-s}(\mathbb{R}^{n})}^{\frac{n-1}{n-s}}\|\mathbb{A}[D]u\|_{L^{1}(\mathbb{R}^{n},d\mathscr{L}^{n})}$$ hold for vector fields $u\in C^{\infty}(\mathbb{R}^{n};\mathbb{R}^{N})$ if and only if the $k$-th order homogeneous linear differential operator $\mathbb{A}[D]$ on $\mathbb{R}^{n}$ is elliptic and cancelling, provided that $s<1$, and give partial results for $s=1$, where stronger conditions on $\mathbb{A}[D]$ are necessary. Here, $\|\mu\|_{L^{1,\lambda}}$ denotes the $(1,\lambda)$-Morrey norm of the measure $\mu$, so that such traces can be taken, for example, with respect to the Hausdorff measure $\mathscr{H}^{n-s}$ restricted to fractals of codimension $0<s<1$. The above class of inequalities give a systematic generalisation of Adams' trace inequalities to the limit case $p=1$ and can be used to prove trace embeddings for functions of bounded $\mathbb{A}$-variation, thereby comprising Sobolev functions and functions of bounded variation or deformation. We moreover establish a multiplicative version of the above inequality, which implies ($\mathbb{A}$-)strict continuity of the associated trace operators on $\text{BV}^{\mathbb{A}}$.

Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms

<p style='text-indent:20px;'>In this paper, we prove two new improved Sobolev inequalities involving weighted Morrey norms in <inline-formula><tex-math id="M1">\begin{document}$ {\dot{H}}^s( \mathbb{R}^{n})\times {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M2">\begin{document}$ {D}^{1, p}( \mathbb{R}^{n})\times{D}^{1, p}( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula>. For instance, the corresponding inequality in <inline-formula><tex-math id="M3">\begin{document}$ {\dot{H}}^s( \mathbb{R}^{n})\!\times\! {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> states that: there exists <inline-formula><tex-math id="M4">\begin{document}$ C\! = \!C(n, s, \alpha, \eta_1, \eta_2)\!&gt;\!0 $\end{document}</tex-math></inline-formula> such that for each <inline-formula><tex-math id="M5">\begin{document}$ (u, v) \!\in\! {\dot{H}}^s( \mathbb{R}^{n})\!\times\! {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ p\!\in\![2, 2^*_{s}(\alpha)) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \theta \!\in\! (\bar{\theta}, \frac{2\eta_1}{2^*_{s}(\alpha)}) $\end{document}</tex-math></inline-formula>, it holds that <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \Big( \int_{ \mathbb{R}^{n} } \frac{ |u|^{\eta_1} |v|^{\eta_2} } { |y|^{\alpha} } dy \Big)^{ \frac{1}{ 2^*_{s} (\alpha) }} \nonumber \\ \!\leq\! C ||u||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}} ||v||_{{\dot{H}}^s(\mathbb{R}^{n})}^{\frac{\theta}{2}+\frac{\eta_2-\eta_1}{2^*_{s} (\alpha)}} ||(uv)||^{\frac{\eta_1}{2^*_{s} (\alpha)}-\frac{\theta}{2}}_{ L^{\frac{p}{2}, \frac{p}{2}(n-2s+r)}(\mathbb{R}^{n}, |y|^{-\frac{p}{2}r}) }, ~~~~(0)$ \end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M8">\begin{document}$ s \!\in\! (0, 1) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M9">\begin{document}$ 0\!&lt;\!\alpha\!&lt;\!2s\!&lt;\!n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \eta_1\!+\!\eta_2\! = \!2^*_{s}(\alpha)\!: = \!\frac{2(n-\alpha)}{n-2s} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ 1\!&lt;\!\eta_1\!\leq\!\eta_2\!&lt;\!\eta_1\!+\!\frac{\alpha}{s} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M12">\begin{document}$ \bar{\theta}\! = \!\max \Big\{ \frac{2}{2^*_{s}(\alpha)}, \frac{2\eta_1}{2^*_{s}(\alpha)} -\frac{2t(\frac{\alpha}{2s}-\frac{\alpha}{n})}{2^*_{s}(\alpha) -\frac{2\alpha}{n}}\Big\} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M13">\begin{document}$ t\! = \!1\!-\!\frac{(\eta_2-\eta_1)s}{\alpha} $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M14">\begin{document}$ r\! = \!\frac{2\alpha}{ 2^*_{s}(\alpha) } $\end{document}</tex-math></inline-formula>. This inequality, together with its counterpart in <inline-formula><tex-math id="M15">\begin{document}$ {D}^{1, p}( \mathbb{R}^{n})\!\times\!{D}^{1, p}( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> extend similar Sobolev inequality in <inline-formula><tex-math id="M16">\begin{document}$ {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> as well as in <inline-formula><tex-math id="M17">\begin{document}$ {D}^{1, p}( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> obtained by G. Palatucci and A. Pisante [Calc. Var., <b>50</b> (2014)] to the product spaces <inline-formula><tex-math id="M18">\begin{document}$ {\dot{H}}^s( \mathbb{R}^{n})\times {\dot{H}}^s( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M19">\begin{document}$ {D}^{1, p}( \mathbb{R}^{n})\times{D}^{1, p}( \mathbb{R}^{n}) $\end{document}</tex-math></inline-formula>, respectively. <p style='text-indent:20px;'>With the help of the inequality (1), we succeed in obtaining some new existence results for doubly critical elliptic systems involving fractional Laplacian and Hardy terms.

A bridge connecting Lebesgue and Morrey spaces via Riesz norms

In this article, via combining Riesz norms with Morrey norms, the authors introduce and study the so-called Riesz–Morrey space, which differs from the John–Nirenberg–Campanato space in subtracting integral means. These spaces provide a bridge connecting both Lebesgue spaces and Morrey spaces which prove to be the endpoint spaces of Riesz–Morrey spaces. Moreover, the authors introduce a block-type space which proves to be the predual space of the Riesz–Morrey space.

Multiplicity and concentration results for fractional Choquard equations: Doubly critical case

In this paper, we consider the following fractional Choquard equation: ε2s(−Δ)su+V(x)u=ε−α(Iα∗F(u))F′(u),x∈RN,where N⩾3, s∈(0,1] and α∈(0,N), ε>0 is a parameter, Iα is the Riesz potential, and F(u)≔12α♯|u|2α♯+12α∗|u|2α∗, where 2α♯=N+αN and 2α∗=N+αN−2s are lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. Firstly, by the refined Sobolev inequality with Morrey norm, we show a generalization of Lions type theorem. Secondly, combining this theorem with variational methods, we show the multiplicity and concentration of positive solutions for above equation. Moreover, the multiplicity and concentration results are obtained in the case where F(u) has the lower critical exponent 2α♯, which remains unsolved in the existing literature.

Ground state solution of p-Laplacian equation with finite many critical nonlinearities

In this paper, we consider the following problem: where is the p-Laplacian operator, is the critical Sobolev exponent, are the Hardy–Littlewood–Sobolev critical upper exponents, the parameters satisfy some assumptions. First, we establish the refined Sobolev inequality with Coulomb norm, and show the corresponding best constant is achieved in by a nonnegative function. Second, by using the refined Sobolev inequality with Coulomb norm, the refined Sobolev inequality with Morrey norm and variational methods, we establish the existence of nonnegative ground state solution for the above problem.

On the invariance of certain vanishing subspaces of Morrey spaces with respect to some classical operators

We consider subspaces of Morrey spaces defined in terms of various vanishing properties of functions. Such subspaces were recently used to describe the closure of $$C_0^\infty ({\mathbb {R}^n})$$ in Morrey norm. We show that these subspaces are invariant with respect to some classical operators of harmonic analysis, such as the Hardy–Littlewood maximal operator, singular type operators and Hardy operators. We also show that the vanishing properties defining those subspaces are preserved under the action of Riesz potential operators and fractional maximal operators.

Equivalence of operator norm for Hardy-Littlewood maximal operators and their truncated operators on Morrey spaces

We will prove that for 1 < p < ∞ and 0 < λ < n, the central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mγc equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. When p = 1 and 0 < λ < n, it turns out that the weak central Morrey norm of the truncated centered Hardy-Littlewood maximal operator Mγc equals that of the centered Hardy-Littlewood maximal operator for all 0 < γ < +∞. Moreover, the same results are true for the truncated uncentered Hardy-Littlewood maximal operator. Our work extends the previous results of Lebesgue spaces to Morrey spaces.

Fractional Operators on Morrey—Lorentz Spaces and the Olsen Inequality

Under the Morrey norm, the fractional integral operator and the fractional maximal operator behave similarly as was initially proved by Adams and Xiao. Later on, Tanaka extended this result. The goal of this note is to extend their results on Morrey–Lorentz norm for exponents of full range. A passage to the vector-valued setting is done. Moreover, an Olsen-type inequality is obtained.

Maximal commutators and commutators of potential operators in new vanishing Morrey spaces

We study mapping properties of commutators in certain vanishing subspaces of Morrey spaces, which were recently used to solve the delicate problem of describing the closure of nice functions in Morrey norm. We show that the vanishing properties defining those subspaces are preserved under the action of maximal commutators and commutators of fractional integral operators.

New result for nonlinear Choquard equations: Doubly critical case

In this paper, we consider the following equation: −Δu+u=(Iα∗F(u))F′(u)inRN,where N⩾5, α∈(0,N−4), and Iα is the Riesz potential, and F(u)≔12α♯|u|2α♯+12α∗|u|2α∗, where 2α♯=N+αN and 2α∗=N+αN−2 are lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. By using the refined Sobolev inequality with Morrey norm and variational methods, we establish the existence of nonnegative solution for above equation. Our result extends the result in Seok (2018).

Global radial solutions in classical Keller–Segel model of chemotaxis

We consider the simplest parabolic-elliptic model of chemotaxis in the whole space in several dimensions. Criteria for the existence of radially symmetric global-in-time solutions in terms of suitable Morrey norms are derived.