fa.functional analysis – short-time Fourier transform of $ f ast g (y) – f ast g (x) $?

To let $ f, g in mathcal {S} ( mathbb R) $, Then the convolution of two Schwartz class functions is again Schwartz class functions, that is $ f ast g in mathcal {S} ( mathbb R). $

It is also known that the Fourier transform by convolution requires pointwise 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 can remember it
$$ widehat {H} ( xi, eta) = has {K} ( xi) has {h} ( xi) delta_0 ( eta) – has {K} ( eta) has {h} ( eta) delta_0 ( xi), $$
can not be integrated.

I am interested in the short-term Fourier transformation of $ G $ in terms of some $ phi in mathcal {S} ( mathbb R ^ 2), $ This 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 is: (i) Is there a way to calculate? $ 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 $?