Abstract In this paper, $$L^p(\mathbb {R}^d,\gamma _\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -boundedness properties for Littlewood–Paley g-functions involving time and spatial derivatives of Ornstein–Uhlenbeck semigroups are established. Here, $$\gamma _\infty$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> </mml:math> denotes the invariant measure. To prove the strong type results for $$1&lt;p&lt; {\infty}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math> , we use R -boundedness. The weak type (1,1) property is established by studying separately global and local operators defined for the Littlewood–Paley g-functions. By the way $$L^p(\mathbb {R}^d,\gamma _\infty )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>γ</mml:mi> <mml:mi>∞</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> -boundedness properties for maximal and variation operators for Ornstein–Uhlenbeck semigroups are proved.

In this paper, we study the boundedness of global continuous linear operators on smooth manifolds. Using the notion of a global symbol, we extend a classical condition of Hörmander type to guarantee the Lp-Lq-boundedness of global operators. Our approach links the mapping properties of continuous linear operators on smooth manifolds with the Lp-estimates of eigenfunctions of operators including a variety of examples, harmonic oscillators, anharmonic oscillators, etc. First, we investigate Lp-boundedness of pseudo-multipliers in the setting of Hörmander–Mihlin type conditions. We also prove L∞-BMO estimates for pseudo-multipliers. Later, we concentrate our investigation to settle Lp-Lq boundedness of the Fourier multipliers and pseudo-multipliers operators for the range 1<p≤2≤q<∞. On the way to achieve our goal of Lp-Lq boundedness, we prove two classical inequalities, namely, Paley inequality and Hausdorff–Young–Paley inequality for smooth manifolds. Finally, we present some examples about the well-posedness of abstract non-linear equations.

In this paper, we investigate properties of hyponormal Toeplitz operators whose symbols are analytic or co-analytic on a Newton space. We establish both necessary and sufficient conditions for a Toeplitz operator Tφ to be hyponormal or normal. Additionally, we give some applications of such results.