In this paper, we establish some Hermite-Hadamard type inequalities for h- convex function on high-dimensional balls and ellipsoids, which extend some known results. Some mappings connected with these inequalities and related results are also obtained.

In this note, we consider the Fourier integral operator T ϕ , a ⁢ f ⁢ ( x ) = ∫ R n e i ⁢ ϕ ⁢ ( x , ξ ) ⁢ a ⁢ ( x , ξ ) ⁢ f ^ ⁢ ( ξ ) ⁢ d ξ with 𝑎 in the forbidden Hörmander class S ρ , 1 m and ϕ ∈ Φ 2 satisfying the strong non-degeneracy condition. For 0 ≤ ρ ≤ 1 , set m ⁢ ( n , ρ , p ) = − ( n − ρ ) ⁢ ( 1 2 − 1 p ) + n p ⁢ ( ρ − 1 ) . When 2 ≤ p ≤ ∞ , 1 ≤ q ≤ ∞ and s > m − m ⁢ ( n , ρ , p ) , we show that T ϕ , a is bounded from the Besov space B p , q s to B p , q s − m + m ⁢ ( n , ρ , p ) . This result is a generalization of some theorems proved by Stein, Meyer, Runst and Bourdaud for the pseudo-differential operator T a with a ∈ S 1 , 1 m , and indices 𝑠 and m ⁢ ( n , ρ , p ) are sharp in some cases.

In this article, we study some Hermite-Hadamard-type inequalities for strongly h -convex functions on co-ordinates in R n , which extend some known results. Some mappings connected with these inequalities and related applications are also obtained.