A capacitance hat used on a shortened antenna ($\le$ 1/8 λ for verticals and $\le$ 1/4 λ for dipoles) converts a largely triangular current distribution with the maximum current at the feedpoint, to a more uniform current distribution or alternatively, it lessens the peak current in the antenna element. This has the effect of:
1.) Raising the Rr (radiation resistance) of the antenna and thus improves the efficiency of the antenna.
2.) It helps to neutralize the inherent capacitive reactance of the shortened antenna and thus generally makes matching the antenna to the feedline an easier task.
Like all shortened antennas, the bandwidth of the antenna is typically less than a "full size" version of the antenna. But the hat will generally broaden the bandwidth compared to other methods of matching the antenna.
The concept of capacity hats on dipoles is not new. Here is a picture of a commercial dipole from DX Engineering. Note the clearly visible wire hats:
Here is a photo from DJ0IP showing his 40 meter cap hat dipole below his spider beam:
The classic modeling approach is to treat the antenna (or in the case of a dipole, each leg) as an open ended transmission line in order to calculate the impedance magnitude of the antenna (or the leg). The degree of wavelength of the antenna (or leg) is also computed. These values allow the reactance that is required to make up the "missing degrees" of the antenna element. Finally, the capacitive reactance required in the capacitance hat is computed. The following universal reactance curve for an open or shorted transmission line is used to compute the reactance required based on the missing degrees:
source: QST, September 1978, Designing a Vertical Antenna by Walter Schulz K3OQF
Inspection of the above diagram shows that if the element is a full 90° (1/4 wavelength) in length, then the transmission line effect will transform the infinite impedance of the open end to a near zero impedance (with near zero reactance) on the opposite end. For any electrical length in-between, the ratio of capacitive reactance to impedance is depicted. If the antenna is of simple wire construction, an impedance calculator such as https://chemandy.com/calculators/round-wire-impedance-calculator.htm may prove sufficient to determine the initial impedance magnitude of the wire. The commonly used formula for calculating impedance magnitude is:
$$Z_{MAG}=60(ln\left(\frac{2L}{r}\right) -1) \tag 1$$
where L is the length of the antenna or the leg of a dipole and r is the radius of the conductor, both in the same units.
The capacitance of the capacitive hat can be approximated from:
$$C_{pF}\approx0.89D \tag 2$$
where D is the diameter of the capacitance hat in inches.
Equation 2 is a reasonable approximation for circular capacitance hats constructed with plates, meshes or skeletal elements (e.g. spokes with a circumferential wire). It is important that the radius of the capacitance hat be <0.1 wavelengths so that the hat does not become a radiating element rather than strictly a capacitor at the end(s) of the antenna.
The capacitive reactance of the hat in ohms is given by the standard capacitive reactance formula:
$$X_C=\frac{1}{2\pi fC} \tag 3$$
where f is the frequency in hertz and C is the capacitance in farads.