arXiv subjects
Publications and source records attributed to Hitoshi Tanaka.
We shall verify the Kakeya (Nikodym) maximal operator $K_{N}$, $N\gg 1$, is bounded on the variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^2)$ when the exponent function $p(\cdot)$ is $N$-modified locally log-Hölder continuous and log-Hölder continuous at infinity.
A description of all the admissible weights similar to the Muckenhoupt class $A_p$ is an open problem for the weighted Morrey spaces. In this paper necessary condition and sufficient condition for two-weight norm inequalities on Morrey spaces to hold are given for the Hardy-Littlewood maximal operator. Necessary and sufficient condition is also verified for the power weights.
Let $σ_i$, $i=1,2,3$, denote positive Borel measures on $\mathbb{R}^n$, let $\mathcal{D}$ denote the usual collection of dyadic cubes in $\mathbb{R}^n$ and let $K:\,\mathcal{D}\to[0,\infty)$ be a map. In this paper we give a characterization of the trilinear embedding theorem. That is, we give a characterization of the inequality $$ \sum_{Q\in\mathcal{D}} K(Q)\prod_{i=1}^3\left|\int_{Q}f_i\,dσ_i\right| \le C \prod_{i=1}^3 \|f_i\|_{L^{p_i}(dσ_i)} $$ in terms of discrete Wolff's potential and Sawyer's checking condition, when $1<p_1,p_2,p_3<\infty$ and $\frac1{p_1}+\frac1{p_2}+\frac1{p_3}\ge 1$.
Two weight trace inequalities for positive dyadic operators are characterized in terms of discrete Wolff's potentials in the upper triangle case.
In a filtered measure space, a characterization of weights for which the trace inequality of a positive operator holds is given by the use of discrete Wolff's potential. A refinement of the Carleson embedding theorem is also introduced. Sawyer type characterization of weights for which a two-weight norm inequality for a generalized Doob's maximal operator holds is established by an application of our Carleson embedding theorem. Moreover, Hytönen-Pérez type one-weight norm estimate for Doob's maximal operator is obtained by the use of our two-weight characterization.
Sediment transport over wave-induced ripples is a very complex phenomenon where available models fail to provide accurate predictions. For coastal engineering applications, the 1-DV advection-diffusion equation could be used with an additional parameter α related to the process of vortex shedding above ripples (Absi, 2010). The aim of this study is to provide simple practical analytical tools. An analytical eddy viscosity profile was validated by DNS data of turbulent channel flows (Absi et al., 2011). In this study, we will show that: (1) the period-averaged eddy viscosity in oscillatory boundary layers could be described by this simple analytical formulation; (2) The shape of the vertical profile is validated by period-averaged eddy viscosity of baseline (BSL) k-ω model (Suntoyo and Tanaka, 2009) for sinusoidal and asymmetric waves; (3) The vertical eddy viscosity profile depends on the wave non-linearity parameter and requires therefore a specific calibration.