Let $f, g\in \mathcal{S}(\mathbb R)$. Then the convolution of two Schwartz class function is again Schwartz class function, that is, $f\ast g \in \mathcal{S}(\mathbb R).$
It is also known that the Fourier transform takes convolution to point wise multiplication.
Now we define
$$H(x,y)= \int_{x}^{y} \frac{d}{dt}(f\ast g)(t) dt = f\ast g (y)- f\ast g(x), \ (x, y \in \mathbb R)$$
We may, formally, notice $$\widehat{H}(\xi, \eta)= \hat{K}(\xi) \hat{h}(\xi) \delta_0(\eta)- \hat{K}(\eta) \hat{h}(\eta) \delta_0(\xi),$$ may not be integrable.
I'm interested in the the short-time Fourier transform of $G$ with respect to some $\phi \in \mathcal{S}(\mathbb R^2),$ that is, \begin{eqnarray*} V_{\phi}H(\bar{x}, \bar{w}) & = & \int_{\mathbb R^{2}} H(t)\phi(t-\bar{x}) e^{-2\pi i \bar{w}\cdot t} dt, \ \ (\bar{x}, \bar{y} \in \mathbb R^2). \end{eqnarray*}
My Question are: (I) Is there any way to compute $V_{\phi} H$? (2) Can we expect $ \left\|\|V_{\phi}G(\bar{x}, \bar{w})\|_{L^{\infty}_{\bar{x}}(\mathbb R^2)}\right\|_{L^{1}_{\bar{w}}(\mathbb R^2)} < \infty$?