We present a maximal Lq(Lp)-regularity theory with Muckenhoupt weights for the equationHere, ∂ α t is the Caputo fractional derivative of order α ∈ (0, 2) and a ij are functions of (t, x). Precisely, we show thatwhere 1 < p, q < ∞, γ ∈ R, and w 1 and w 2 are Muckenhoupt weights, and this implies that we prove maximal regularity theory. Our approach is based on Fourier multiplier and singular integral theory with Muckenhoupt weights. Also, we use abstract interpolation theory to find interpolation inequality in weighted Sobolev space.

We investigate an Lq(Lp)-regularity (1 < p, q < ∞) theory for space-time nonlocal equations with zero initial conditionHere, ∂ α t is the Caputo fractional derivative of order α ∈ (0, 1) and L is an integro-differential operator|y|)dy 2020 Mathematics Subject Classification. 35B65, 26A33, 47G20.

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We deal with the Sobolev space theory for the stochastic partial differential equation (SPDE) driven by Wiener processeswith some constants κ 0 > 0 and δ 0 ∈ (0, 1]. We prove the uniqueness and existence results in Sobolev spaces, and obtain the maximal regularity results of solutions.

The electronic structures and two-terminal magnetoconductance of a hybrid quantum ring are studied. The backscattering due to energy-resonance is considered in the conductance calculation. The hybrid magnetic-electric quantum ring is fabricated by applying an antidot electrostatic potential in the middle of a magnetic quantum dot. Electrons are both magnetically and electrically confined in the plane. The antidot potential repelling electrons from the center of the dot plays a critical role in the energy spectra and magnetoconductance. The angular momentum transition in the energy dispersion and the magnetoconductance behavior are investigated in consideration of the antidot potential variation. Results are shown using a comparison of the results of the conventional magnetic quantum dot.