Weighted estimates for bilinear square functions with non-smooth kernels and commutators

A Singular Tempered Sub-Diffusion Fractional Model Involving a Non-Symmetrically Quasi-Homogeneous Operator

“…where 0 < α ≤ 1, 1 < β ≤ 2. In the past twenty years, there has been a growing interest in the profit derived from advancing space theories [18][19][20], regular theories [21][22][23][24][25], operator methods [26][27][28][29][30], iterative techniques [31][32][33], the moving sphere method [34], critical point theories [35][36][37][38], and tempered fractional calculus. This surge in attention has not only propelled the rapid progress of these disciplines, but has also spurred corresponding contributions across various fields.…”

The Iterative Properties for Positive Solutions of a Tempered Fractional Equation

“…Benefiting from the development of the nonlinear analysis theories, in recent years, many powerful tools, such as function spaces theories [35][36][37][38][39][40][41], regularity theories [42][43][44][45][46], the operator technique [47][48][49][50][51][52], the method of upper and lower solutions [53,54], the method of moving sphere [55], variational theories [56][57][58][59][60], and so on, have been developed to solve various partial or ordinary differential equations. Thus, inspired by the above results, in this paper, we investigate the iterative properties of positive solutions for Equation (1) by using the space and operator theory as well as some analytical techniques.…”

The extreme solutions for a σ‐Hessian equation with a nonlinear operator

“…Various nonlinear analysis theories and methods may be used to study the 𝜎-Hessian equation, such as the spaces theories [12][13][14][15][16][17][18][19][20][21], smoothness theories [22][23][24][25][26][27], operator theories [28][29][30][31], fixed point theorems [32][33][34][35][36], sub-super solution methods [37][38][39], monotone iterative techniques [40,41], and the variational method [42][43][44]. For example, by adopting the sub-super solution method, Zhang et al [37] recently established the interval of the eigenvalue in which the existence of solutions for the following singular augmented 𝜎-Hessian equation is guaranteed…”

“…Various nonlinear analysis theories and methods may be used to study the $$$ \sigma $$$‐Hessian equation, such as the spaces theories [12–21], smoothness theories [22–27], operator theories [28–31], fixed point theorems [32–36], sub‐super solution methods [37–39], monotone iterative techniques [40, 41], and the variational method [42–44]. For example, by adopting the sub‐super solution method, Zhang et al [37] recently established the interval of the eigenvalue in which the existence of solutions for the following singular augmented $$$ \sigma $$$‐Hessian equation is guaranteed where $$$ {B}_1&amp;amp;#x0003D;\left\{x\in {\mathrm{\mathbb{R}}}&amp;amp;#x0005E;M:&amp;amp;#x0007C;x&#x0007C;&amp;lt;1\right\} $$$, …”