Test Schwartz functions at $p$ for theta lifts of Hida families

We construct Hida families of theta lifts from definite orthogonal and unitary groups. A major ingredient of the construction is the choice of test Schwartz functions at places dividing $p$. We select a special type of Schwartz functions endowed with the equivariance property for the action of ${\mathbb{U}}_p$-operators.